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On the Einstein-Weyl and conformal self-duality
equations
M. Dunajski, E.V. Ferapontov and B. Kruglikov
DAMTP, Centre for Mathematical Sciences,
University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, United Kingdom;
and
Department of Computer Science,
Faculty of Physics and Applied Informatics,
University of Lodz,
Pomorska 149/153, 90-236 Lodz, Poland;
Department of Mathematical Sciences
Loughborough University, Loughborough,
Leicestershire LE11 3TU, United Kingdom;
Institute of Mathematics and Statistics, NT-Faculty,
University of Tromsø, Tromsø 90-37, Norway
e-mails:
M.Dunajski@damtp.cam.ac.uk
E.V.Ferapontov@lboro.ac.uk
boris.kruglikov@uit.no
Abstract
The equations governing anti-self-dual and Einstein-Weyl conformal geometries
can be regarded as ‘master dispersionless systems’ in four and three dimensions
respectively. Their integrability by twistor methods has been established by Penrose
and Hitchin. In this note we present, in specially adapted coordinate systems,
explicit forms of the corresponding equations and their Lax pairs. In particular,
we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by
a solution to the Manakov-Santini system, and we find a system of two coupled
third-order scalar PDEs for a general anti-self-dual conformal structure in neutral
signature.
1
1 Introduction
There exist two key ‘integrable’ conformal geometries, namely Einstein-Weyl geometry
in three dimensions, and anti-self-dual (ASD) geometry in four dimensions (see [7] for
a comprehensive overview). In spite of their fundamental role in twistor theory, general
relativity and the theory of dispersionless integrable systems, these geometries remain
largely unknown to the integrable systems community due to a lack of explicit coordinate
formulae for the underlying PDEs and Lax pairs. The aim of this paper is to present
them in their simplest possible forms, in specially adapted coordinates.
In Section 2 we discuss Einstein-Weyl (EW) structures in three dimensions. Recall that
an EW geometry on a three-dimensional manifold M3 consists of a conformal structure
[g] and a symmetric connection D compatible with [g] in the sense that, for any g ∈ [g],
Dg = ω ⊗ g
for some covector ω, and such that the trace-free part of the symmetrized Ricci tensor of D
vanishes. Using Cartan’s approach relating EW structures to a special class of third-order
ODEs we shall demonstrate the following
Theorem 1 There exists a local coordinate system (x, y, t) onM3 such that any Lorentzian
Einstein–Weyl structure is locally of the form
g = −(dy − vxdt)
2 + 4(dx− (u− vy)dt)dt, ω = −vxxdy + (4ux − 2vxy + vxvxx)dt, (1)
where the functions u and v on M3 satisfy a coupled system of second-order PDEs
P (u) + u2x = 0, P (v) = 0, (2)
where
P = ∂x∂t − ∂
2
y + (u− vy)∂x
2 + vx∂x∂y.
System (2) is known as the Manakov-Santini system, and was originally derived in [31] as
a two-component generalization of the dispersionless KP equation. It was shown in [16]
that any solution to (2) gives rise to an EW structure of the form (1), but the question
whether all EW structures arise in that way has remained open. System (2) possesses the
Lax representation [X1, X2] = 0 where
X1 = ∂y − (λ+ vx)∂x − ux∂λ, X2 = ∂t − (λ
2 + vxλ− u+ vy)∂x − (uxλ+ uy)∂λ
are vector fields on the correspondence space M3 × RP 1, where λ ∈ RP 1. Projecting
integral surfaces of the distribution spanned by X1, X2 from M × RP
1 to M3 yields
a two-parameter family of surfaces in M3 which are null with respect to the conformal
structure [g], and totally geodesic in the Weyl connection D (the existence of such surfaces
is equivalent to the EW property [9]). System (2) consists of 2 second-order PDEs for
2 functions of 3 independent variables, and its general solution in real-analytic category
depends on 4 arbitrary functions of 2 variables: this confirms Cartan’s count [9].
2
The relation between EW geometry and dispersionless integrable systems in three
dimensions has been known since [39, 17, 16]. It was observed recently in [22] that the
dispersionless integrability of various classes of second-order PDEs is equivalent to the
EW property of conformal structures defined by their principal symbols. Moreover, in
many examples the covector ω is expressed in terms of g ∈ [g] by the universal explicit
formula
ωk = 2gkjDxs(g
js) +Dxk(ln det gij). (3)
Here Dxs denotes total derivative with respect to x
s, and (x1, x2, x3) = (x, y, t). In three
dimensions this formula is invariant under the transformation
g → ϕ2g, ω → ω + 2d lnϕ, where ϕ :M3 → R+, (4)
that keeps the Einstein-Weyl equations invariant. The Manakov-Santini system fits into
this framework: the covector in (1) is given by formula (3), and the principal symbol of
system (2) equals P 2 (it is doubly degenerate) where P , viewed as a symmetric bivector,
gives rise to the EW metric g given by (1).
In Section 3 we study ASD conformal structures in four dimensions. Recall that a
conformal structure [g] is called anti-self-dual if the self-dual part of the Weyl tensor of
any g ∈ [g] vanishes: W+ =
1
2
(W + ∗W ) = 0. We shall establish the following
Theorem 2 There exist local coordinates (w, z, x, y) such that any ASD conformal struc-
ture in signature (2, 2) is locally represented by a metric
g = dwdx+ dzdy + Fydw
2 − (Fx +Gy)dwdz +Gxdz
2, (5)
where the functions F,G : M4 → R satisfy a coupled system of third-order PDEs,
∂x(Q(F ))−∂y(Q(G)) = 0, (∂w−Fy∂x+Gy∂y)Q(G)+(∂z+Fx∂x−Gx∂y)Q(F ) = 0, (6)
where
Q = ∂w∂x + ∂z∂y − Fy∂x
2 −Gx∂y
2 + (Fx +Gy)∂x∂y.
System (6) arises as [X1, X2] = 0 from the dispersionless Lax pair
X1 = ∂w −Fy∂x+Gy∂y + λ∂y +Q(F )∂λ, X2 = ∂z +Fx∂x −Gx∂y + λ∂x −Q(G)∂λ. (7)
Projecting integral surfaces of the distribution spanned by X1, X2 in the correspondence
space M4 × RP 1 to M4 gives a 3-parameter family of totally null surfaces with self-dual
tangent bi-vector. These are the α-surfaces of the corresponding conformal structure [g].
The existence of such surfaces is equivalent to the ASD property [34]. System (6) consists
of 2 third-order PDEs for 2 functions of 4 independent variables, and its general solution
depends on 6 arbitrary functions of 3 variables: this agrees with the count of [25] based
on the Cartan-Ka¨hler theory. The Lax pair (7) has appeared in [1], where the Riemann–
Hilbert problem associated to (7) has been formulated. The fact that the system of 3
second order PDEs derived in [1, 2] leads to system (6) has been recently pointed out in
[41].
3
ASD equations and their reductions provide a number of key examples of dispersion-
less integrable systems in four dimensions [35, 15, 19]. It was conjectured in [22] that the
dispersionless integrability of some four-dimensional PDEs is equivalent to the require-
ment that the principal symbol of the equation defines a conformal structure that must
be ASD on every solution. In [22] this was demonstrated to be the case for integrable
symplectic Monge-Ampe`re equations [14]. The example of ASD equations fits into this
scheme: the principal symbol of system (6) equals Q3 (it is triply degenerate), and the
symmetric bivector Q gives rise to the ASD metric g given by (5).
Reality conditions. If all coordinates and functions in Theorems 1 and 2 are assumed
to be real, then the corresponding conformal structures in three and four dimensions have
Lorentizan or (2, 2) (also called neutral or Kleinian) signatures, respectively. Alternatively,
one can assume real analyticity and work in the complexified settings where all structures
are assumed to be holomorphic. We shall make this additional assumption whenever we
rely on the Cauchy–Kowalevskaya theorem to assert that a general solution depends on
m functions of n variables.
Acknowledgements
We are indebted to Leonid Bogdanov, Paolo Santini, and an anonymous referee for point-
ing out that the Lax formulation (7) first appeared in [1].
2 Einstein-Weyl geometry
The twistor integrability of EW equations was established in [26]. It was demonstrated
in [17] that EW equations possess a Lax pair given by two vector fields that form an
integrable distribution, and may contain derivatives with respect to the spectral parame-
ter. Integral manifolds of this distribution provide the 2-parameter family of null totally
geodesic surfaces, and, as shown by Cartan [9], the EW property is equivalent to the
existence of such family. However, the explicit coordinate form of the Lax pair has not
been exhibited in the general case. Below we list various forms of EW equations, as well
as their Lax pairs, in specially adapted coordinates.
2.1 Einstein-Weyl equations in Cartan’s approach
Our proof of Theorem 1 builds on Cartan’s approach to Einstein-Weyl geometry via
special third-order ODEs [10]. We shall briefly review it following the paper of Tod [38].
Consider an equivalence class of third-order ODEs
Y ′′′ = F (X, Y, Y ′, Y ′′), (8)
modulo point transformations (X, Y ) → (X¯(X, Y ), Y¯ (X, Y )). Here ′ = d/dX . Let the
general solution of (8) be of the form
Y = Z(X, xj), (9)
4
where xj are coordinates on the three–dimensional solution space M3. The necessary and
sufficient conditions for the solution space M3 to carry an Einstein–Weyl structure such
that the 2-parameter family of surfaces in M3 corresponding to fixing (X, Y ) in (9) is null
and totally geodesic, are given by the vanishing of the Wunshmann and Cartan invariants
W and C. These invariants are given by
W = 1
6
D2FQ −
1
3
FQDFQ −
1
2
DFP +
2
27
F 3Q +
1
3
FQFP + FY ,
C = (1
3
DFQ −
1
9
F 2Q − FP )FQQ +
2
3
FQFQP − 2FQY + FPP + 2WQ,
where D = ∂X + P∂Y +Q∂P + F∂Q is the total derivative.
The above W is actually a relative contact invariant, while C is a relative point
invariant (so their vanishing is an invariant condition for respective pseudogroups).
Following the approach of Tod [38] (see [32, 30] for other approaches), the conformal
structure and the covector are given by
g = 2 dY dQ− 2
3
FQ dY dP + (
1
3
DFQ −
2
9
F 2Q − FP ) dY
2 − dP 2,
ω = 2
3
(FQP −DFQQ) dY +
2
3
FQQdP,
(10)
where in these expressions X is fixed. Both g and ω depend on X explicitly, but a change
in X corresponds to a gauge transformation of the form (4). Thus, as long asW = C = 0,
the resulting Einstein-Weyl structure is independent of X .
In this approach the EW equations, W = C = 0, constitute an overdetermined system
of two PDEs for a scalar function F of four variables. One can show that this system is
compatible (formally integrable), which follows from the vanishing of the Mayer bracket
[W,C] = 0 [28]. In other words, this system of third- and second-order PDEs is in invo-
lution (after three prolongations). The characteristic variety is a complete intersection,
so the general solution is parametrized by 6 = 3 · 2 functions of 2 variables (we refer to
[11, 6, 29] for the general dimension theory of solution spaces). However, the system is
point invariant, so the diffeomorphism freedom is 2 functions of 2 variables, and hence-
forth the actual solution space is parametrized by 4 = 6−2 functions of 2 variables. This
re-proves Cartan’s count.
Proof of Theorem 1. Setting A = −2
3
FQ, B =
1
3
DFQ −
2
9
F 2Q − FP one can rewrite
(10) in the form
g = 2 dY dQ+ AdY dP +B dY 2 − dP 2,
ω = (AP − 2BQ −
1
2
AAQ) dY −AQdP.
To bring the corresponding EW equations to the desired form we fix X = 0, and set the
variables as follows: Q(0) = x, P (0) = y, Y (0) = 2t, A|X=0 = a, B|X=0 = −b −
1
4
a2,
where now (x, y, t) are local coordinates on M3, and a, b : M3 → R. This results in
g = 4dt dx+ 2a dt dy − (a2 + 4b) dt2 − dy2,
ω = (aax + 2ay + 4bx) dt− axdy.
(11)
5
The EW equations reduce to a pair of second-order conservative PDEs,
(at + aay + bax)x = (ay)y, (bt + bbx − aby)x = (by − 2abx)y, (12)
which coincide with the Manakov-Santini system (2) upon substitution a = vx, b = u−vy ,
see also [33]. Note that system (12) allows one to uniquely reconstruct g and ω in (11):
the conformal structure g comes from the principal symbol of system (12), and ω is given
by formula (3). Since the construction directly follows from Cartan’s approach, we can
conclude that the Manakov-Santini system gives all EW structures. The general solution
of system (12) depends on 4 arbitrary functions of 2 variables which agrees with Cartan’s
result.

The Lax representation of (12) has the form [X1, X2] = 0, where
X1 = ∂t − (λ
2 − aλ− b) ∂x +m∂λ, X2 = ∂y − λ ∂x + n ∂λ,
and
m = −axλ
2 + (aax − ay − bx)λ+ (abx − by), n = −axλ− bx.
We point out that this Lax pair transforms to the one of the Manakov-Santini system
presented in the Introduction via the change of variables a = vx, b = u− vy, λ = λ˜+ vx.
Taking a linear transformation of the Lax vector fields results in a Lax pair linear in the
parameter λ. A further affine translation of λ with non–constant coefficients can be used
to bring the Lax pair to the canonical form used in [17, 16].
Projecting integral surfaces of the distribution spanned by X1 and X2 in the 4D space
with coordinates (x, y, t, λ) to the space M3 with coordinates (x, y, t) one obtains a 2-
parameter family of null totally geodesic surfaces of the corresponding EW structure.
There is an RP 1–worth of such surfaces through any point in M3.
The constraint a = 0, b = u reduces system (12) to the dispersionless KP equation,
(ut + uux)x = uyy, while the corresponding EW structure reduces to the one from [17]:
g = 4dt dx− 4u dt2 − dy2, ω = 4ux dt.
Any EW structure which admits a parallel vector field can locally be put in this form.
Another possible reduction is u = 0. This corresponds to the most general hyper-CR
Einstein–Weyl structure [18].
2.1.1 Translationally non-invariant version of the Manakov-Santini system
Here our starting point is the general ansatz for a metric in the conformal class and a
covector [16]. Using the diffeomorphism and conformal freedom, the representative metric
can be put in the form (11). Set ω = ω1 dt + ω2 dx+ ω3 dy. Imposing the Einstein-Weyl
conditions we obtain a system of 5 PDEs for a, b, ωi, that is not presented here due to its
complexity. The corresponding Lax pair has the form [X1, X2] = 0, where
X1 = ∂t − (λ
2 − aλ− b) ∂x +m∂λ, X2 = ∂y − λ ∂x + n ∂λ,
6
and m and n are the following cubic and quadratic polynomials in λ:
m = −
1
2
ω2λ
3 +
1
4
(aω2 + 4ω3)λ
2 −
1
2
(ω1 + bω2 + 2aω3 − aax − 2bx)λ
+
1
4
(aω1 + abω2 + a
2ω3 − 2aay − 4by),
n = −
1
4
ω2λ
2 +
1
2
(ω3 − ax)λ−
1
4
(ω1 + bω2 + aω3 − 2ay).
One of the five EW equations has the simple form (ω2)x + ω
2
2/2 = 0. This leads to the
natural branching:
Case 1: ω2 = 0. Up to further elementary integration and changes of variables this case
can be reduced to the form (11), with the Manakov-Santini system (12) for a, b.
Case 2: ω2 = 2/x (strictly speaking, ω2 = 2/(x+f(y, t)), however, f(y, t) can be removed
by a transformation x→ x+f(y, t), that preserves the form of the metric after appropriate
re-definition of a and b). This branch can be viewed as a translationally non-invariant
(x-dependent) version of the Manakov-Santini system.
In view of Theorem 1 both branches are equivalent, but we have been unable to find
a combination of a conformal rescaling and a coordinate transformation which reduces
Case 2 to Case 1.
2.2 Einstein-Weyl equations via Bogdanov’s system
The following system was proposed by Bogdanov [3] as a two-component generalization
of the dispersionless Toda equation:
(e−φ)tt = mtφxy −mxφyt, mtte
−φ = mxmyt −mtmxy.
It possesses a Lax representation [X1, X2] = 0, where
X1 = ∂x −
(
λ+
mx
mt
)
∂t + λ
(
φt
mx
mt
− φx
)
∂λ, X2 = ∂y +
1
λ
e−φ
mt
∂t +
(e−φ)t
mt
∂λ.
It was observed in [22] that, for any solution of the Bogdanov system, the metric
g = (mxdx+mtdt)
2 + 4e−φmtdxdy
and the covector
ω =
(
mtt
m2t
− 2
φt
mt
)
(mx dx+mt dt) + 2
myt
mt
dy
satisfy the EW equations. Note that g comes from the principal symbol of the system,
and ω is given by formula (3). The general solution of the Bogdanov system depends on 4
arbitrary functions of 2 variables. It is natural to expect that this gives (locally) a generic
EW structure.
Setting m = t one obtains the SU(∞) Toda equation [4], (e−φ)tt = φxy, while the
corresponding EW structure reduces to the one from [39]:
g = dt2 + 4e−φdxdy, ω = −2φtdt.
7
2.3 Einstein-Weyl equations in diagonal coordinates
Note that any three-dimensional metric possesses diagonal coordinates depending locally
on 3 arbitrary functions of 2 variables [11, 13]. We can therefore use conformal freedom
g → ϕg, ω → ω + d lnϕ to set
g = a2dt2 − dx2 + b2dy2, ω = ω1dx+ ω2dy + ω3dt.
In this case the EW equations give rise to a system of five PDEs for the five functions
a, b, ωi, which are second-order in a, b, and first-order in ωi (this system is not presented
explicitly due to its complexity). It possesses the Lax pair [X1, X2] = 0:
X1 = ∂t − a cosλ ∂x +m ∂λ, X2 = ∂y − b sin λ ∂x + n ∂λ,
where
m = −
a
2b
ω2 sin
2 λ−
1
2
ω3 sinλ cosλ+ (
1
2
aω1 − ax) sinλ+
ay
b
,
n = b
2a
ω3 cos
2 λ+ 1
2
ω2 sinλ cosλ− (
1
2
bω1 − bx) cosλ−
bt
a
.
The general solution of the system for a, b, ωi depends locally on 7 = 2 · 2+ 3 · 1 arbitrary
functions of 2 variables (recall the order of PDEs). Since diagonal coordinates exist with
the freedom of 3 arbitrary functions of 2 variables, this again confirms that EW structures
depend on 4 = 7− 3 arbitrary functions of 2 variables.
The above system possesses a reduction [22]
g = (1− e−u)dt2 − dx2 + (eu − 1)dy2, ω =
eu + 1
eu − 1
uxdx− uydy + utdt,
for which the EW equations reduce to the scalar second-order PDE
uxx + uyy − (ln(e
u − 1))yy − (ln(e
u − 1))tt = 0.
This is the dispersionless limit of the ‘gauge-invariant’ Hirota equation [21].
3 Anti-self-duality equations
A conformal structure g on a four-dimensional manifold is called anti-self-dual (ASD) if
the self-dual (SD) part of its conformal Weyl tensor vanishes: W+ =
1
2
(W + ∗W ) = 0.
The twistor-theoretic integrability of the ASD condition was established in [34]. It was
shown in [25] that generic ASD structure depends on 6 arbitrary functions of 3 variables.
The existence of a Lax pair is implicitly built in the fact that any ASD structure possesses
a 3-parameter family of totally null α-surfaces. Below we present several forms of ASD
equations and their Lax pairs in specially adapted coordinates.
8
3.1 Anti-self-duality equations in Pleban´ski-Robinson coordi-
nates
Here we present explicit formulae, including the corresponding Lax pair, in Pleban´ski-
Robinson coordinates (w, z, x, y) where the metric g in the ASD conformal class on an
open set M4 ⊂ R4 takes the hyper-heavenly form,
g = dwdx+ dzdy + pdw2 + 2qdwdz + rdz2, (13)
where p, q, r are functions of all four variables. We assume that all coordinates are real,
so that the signature of g is (2, 2).
Proposition 3.1 The metric (13) has ASD Weyl tensor if the functions p, q, r satisfy
the system of three second order PDEs :
pxx + 2qxy + ryy = 0,
mx + ny = 0,
mz − qmx − rmy + (qx + ry)m = nw − pnx − qny + (px + qy)n,
(14)
where
m := pz − qw + pqx − qpx + qqy − rpy, n := qz − rw + qry − rqy + prx − qqx. (15)
Conversely, any ASD conformal structure is locally of the form (13), where (p, q, r) satisfy
system (14).
Proof. We will make use of the isomorphisms Λ2+ = S
′ ⊙ S ′ and TM4 = S ⊗ S ′, where
S and S ′ are real rank-two symplectic vector bundles (spin bundles), Λ2+ is the rank-
three bundle of self-dual two-forms on M4 and ⊙ is the symmetrized tensor product. The
seminal result of Penrose [34] asserts that the ASD condition is equivalent to the existence
of a 3-parameter family of α-surfaces (totally null surfaces in M4 with SD tangent bi-
vector). This means that any section of S ′ corresponds to a SD two-form defining a
2D distribution integrable in the Frobenius sense. To arrive at the canonical form (13)
we select a 2-parameter family of α–surfaces corresponding to a section ι ∈ Γ(S ′). Let
Σ ∈ Γ(Λ2+) be a SD two–form corresponding to this section. It is Frobenius-integrable so
there exist independent functions x and y on M4 such that Ker(Σ) = Span(∂/∂x, ∂/∂y).
We can moreover rescale the spinor ι so that the corresponding two–form Σ is closed
and proportional to dw ∧ dz. Therefore w and z are constant on each α–surface in the
2-parameter family, and (x, y) are coordinates on the surface. The α–surfaces are totally
null so that the conformal structure is represented by
g = e00
′
e11
′
− e01
′
e10
′
, (16)
where
e00
′
= adz, e10
′
= bdw, e01
′
= −dx− pdw − qdz, e11
′
= dy + qdw + rdz; (17)
9
here (a, b, p, q, r) are so far unspecified functions (we have set the dzdx and dwdy coeffi-
cients in g to 0 by exploiting the coordinate freedom in the choice of (x, y)). To examine
the ASD condition we choose a basis of S ′ consisting of two spinors (o, ι). The self-dual
Weyl spinor W ′ is a section of Sym4(S ′) given by
W ′ = W0 o o o o+ 4W1 o o o ι+ 6W2 o o ι ι+ 4W3 o ι ι ι+W4 ι ι ι ι,
where the symmetrised tensor product is implicit in this formula. We find that W4
vanishes identically and that W3 =
1
4
∂x∂yln(a/b). Therefore a = b exp (α + β), where
α = α(x, w, z), β = β(y, w, z).
Now we make a coordinate transformation x → x˜(x, w, z), y → y˜(y, w, z) such that
∂y˜/∂y = exp (−β) and ∂x˜/∂x = exp (α). Finally we redefine (p, q, r) and conformally
rescale the resulting metric by b−1 exp (−α). This puts the metric in the form (13), with
the corresponding null tetrad given by (17) with a = b = 1. So far our proof has more or
less followed the construction of Pleban´ski and Robinson [36], but now we shall proceed
differently. Instead of imposing the Einstein equations we shall assume that the remaining
three components of W ′ vanish. This gives the coupled system (14). 
System (14) possesses the Lax pair [X1, X2] = 0, where X1 and X2 are λ-dependent
vector fields,
X1 = ∂w − p∂x − q∂y + λ∂y + [m− λ(px + qy)]∂λ, (18)
X2 = ∂z − q∂x − r∂y − λ∂x + [n− λ(qx + ry)]∂λ,
where m,n are given by expressions (15). This Lax pair is a coordinate realization of the
general twistor distribution LA = (X1, X2) on the projectivized spin bundle S
′ given by
LA = pi
A′eAA′ − pi
A′piB
′
piC
′
ΓAA′B′C′
∂
∂λ
, (19)
where the indices A,B,A′, B′, . . . take values 0 or 1, the vector fields eAA′ are dual to
the one forms (17), ΓAA′B′C′ are components of the spin connection, and pi
A′ = (1, λ) are
homogeneous coordinates on the fibres of PS ′. Projecting integral surfaces of the distri-
bution spanned by X1 and X2 from the correspondence space M
4×RP 1 with coordinates
(w, z, x, y, λ) to M4, we obtain a 3-parameter family of null surfaces (α-surfaces) of the
conformal structure g. The spectral parameter λ on RP 1 is a coordinate on the circle of
α–surfaces at each point ofM4. The conformal structure (13) can be read off the principal
symbol of system (14). Indeed, the principal symbol of (14) equals Q3, where
Q = ∂w∂x + ∂z∂y − p ∂
2
x − 2q ∂x∂y − r ∂
2
y ,
and the inverse matrix of the symmetric bivector Q defines conformal structure (13).
Theorem (2) states that a further simplification is possible, so that ASD conditions
reduce to a system of 2 third-order PDEs for 2 functions. The proof below uses one of
the equations from Proposition (3.1) as integrability conditions.
Proof of Theorem 2. Rewrite the first equation in (14) as (px + qy)x + (qx + ry)y = 0,
which implies the existence of a function s such that px = sy−qy and ry = −sx−qx. These
10
two equations can again be regarded as the integrability conditions for the existence of
two functions F , and G on M4 such that
p = Fy, q = −(Fx +Gy)/2, r = Gx.
The remaining two equations in (14) now yield (6).
To exhibit a simple Lax pair for (6) we shall make a linear transformation (null rota-
tion) of the frame of S ′ which does not change metric (16):
e11
′
→ e11
′
+ γe10
′
= dy −Gydw +Gxdz, e
01′ → e01
′
+ γe00
′
= −dx− Fydw + Fxdz,
here γ = (Fx −Gy)/2. In this spin frame, Lax pair (19) gives (7). 
3.2 Anti-self-duality equations and torsion-free ODE systems
In the spirit of Cartan, it was shown by Grossman [25] that there is a one-to-one correspon-
dence between ASD conformal structures in signature (2, 2) and systems of second-order
ODEs with vanishing generalized Wilczynski invariants (torsion-free systems in his termi-
nology). In particular, Grossman has shown that a generic torsion-free system depends
on 6 arbitrary functions of 3 variables. The canonical form (5) of the ASD metric can be
directly derived from Grossman’s approach [8]: if a torsion-free system of 2 ODEs is of
the form
W ′′ = G(X,W,Z,W ′, Z ′), Z ′′ = F (X,W,Z,W ′, Z ′),
(here prime denotes differentiation byX), then the solution spaceM4 can be parametrized
by fixing X , say X = 0, and defining (w, z, x, y) to be the initial conditions: w =
W (0), z = Z(0), y = W ′(0), x = −Z ′(0). The conformal ASD structure on M4 is
then defined by demanding that points in the (X,W,Z) space correspond to totally null
α-surfaces in M4. In the chosen coordinates this leads to the formula (5), where F,G are
evaluated at X = 0, see [8] for details of this construction.
3.3 Anti-self-duality equations in doubly biorthogonal coordi-
nates
It was demonstrated in [24] that any (analytic) four-dimensional metric can be brought
into block-diagonal form,
g =


a1 a2 0 0
a2 a3 0 0
0 0 b1 b2
0 0 b2 b3

 .
Coordinates of this type are known as doubly biorthogonal. They depend locally on 4
arbitrary functions of 3 variables. Using the conformal freedom to set det g = 1, one
can show that the equations of self-duality reduce to a (complicated) system of 5 second-
order PDEs for the 5 (independent) functions among ai and bi. The general solution of
this system depends locally on 10 arbitrary functions of 3 variables. Since biorthogonal
11
coordinates exist with the freedom of 4 arbitrary functions of 3 variables, this is again in
agreement with the fact that self-dual structures depend locally on 10 − 4 = 6 arbitrary
functions of 3 variables.
3.4 Reductions of self-duality equations
ASD equations possess several geometric reductions of interest:
Hyper-Hermitian case: this case is characterized by the existence of a Lax pair that
does not contain derivatives with respect to the spectral parameter [15, 7]. Taking into
account (7) this leads to a pair of second-order PDEs
Q(F ) = 0, Q(G) = 0. (20)
This system was first derived in [23], where the corresponding conformal structures were
referred to as ‘weak heavenly spaces’. The dispersionless Lax pair for system (20) is
X1 = ∂w − Fy∂x +Gy∂y + λ∂y, X2 = ∂z + Fx∂x −Gx∂y − λ∂x.
In [15] it was shown that all (pseudo) hyper-Hermitian conformal structures locally arise
from solutions to (20). The general solution to this system depends on 4 arbitrary func-
tions of 3 variables. In the special case where F = θy, G = θx, the hyper-Hermitian system
reduces to Plebanski’s 2nd heavenly equation,
θyz + θxw + θ
2
xy − θxxθyy = 0, (21)
and the metric g is Ricci–flat. It depends on 2 arbitrary functions of 3 variables.
Null Ka¨hler case: the ansatz F = θy, G = θx reduces ASD equations (6) to a single
fourth-order PDE for θ [19]:
Q(f) = 0, f = θyz + θxw + θ
2
xy − θxxθyy,
where Q = ∂w∂x + ∂z∂y − θyy∂x
2 − θxx∂y
2 + 2θxy∂x∂y.
(22)
In this case the self-dual two form Σ = dw∧dz corresponding to the two-parameter family
of α–surfaces from the proof of Theorem 3.1 is covariantly constant. Conversely, it was
demonstrated in [19] that any ASD metric g that admits a self-dual covariantly constant
two-form Σ such that Σ∧Σ = 0, is locally given by a solution to (22). The dispersionless
Lax pair for (22) is
X1 = ∂w − θyy∂x + θxy∂y + λ∂y + fy∂λ,
X2 = ∂z + θxy∂x − θxx∂y − λ∂x − fx∂λ.
In the special case f = 0 we recover the second heavenly equation (21).
Other reductions: The coordinate system introduced in Proposition 3.1 is adapted
to a choice of a preferred two–parameter family of α–surfaces determined by a section
ι ∈ Γ(S ′), or equivalently by a Frobenius-integrable simple two form Σ. There are other
12
possibilities which single out a non-degenerate two form Σ such that Σ ∧ Σ 6= 0. This
requires a choice of two independent sections of S ′ and leads to PDEs generalising Pleban-
ski’s 1st heavenly equation [35, 5]. In particular, the Przanowski equation [37] describing
all ASD Einstein metrics with non-vanishing cosmological constant, is written down in
such coordinates. A Lax pair for this equation has recently been found in [27]. Its 2nd
heavenly form analogous to (21) has been given in [12].
References
[1] L. V. Bogdanov, V. S. Dryuma and S. V. Manakov, Dunajski generalization of the second heavenly
equation: dressing method and the hierarchy, J. Phys. A: Math. Theor. 40 (2007) 14383-14393.
[2] L.V. Bogdanov, ‘Interpolating’ differential reductions of multidimensional integrable hierarchies ,
arXiv:1011.0631 (2010).
[3] L.V. Bogdanov, Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy, J. Phys. A
43, no. 43 (2010) 434008, 8 pp.
[4] C. P. Boyer, D. Finley, Killing vectors in self-dual, Euclidean Einstein spaces, J. Math. Phys. 23,
1126–1130.
[5] C. P. Boyer, J. F. Plebanski, Conformally self-dual spaces and Maxwell’s equations, Phys. Lett.
A106 (1984), 125–129.
[6] R. L. Bryant, S. S. Chern, R.B. Gardner, H. L. Goldschmidt, P.A. Griffiths, Exterior differential
systems, MSRI Publications 18, Springer-Verlag (1991).
[7] D.M.J. Calderbank, Integrable background geometries, SIGMA 10 (2014) 034. Special Issue on
Progress in Twistor Theory.
[8] S. Casey, M. Dunajski, and K. P. Tod, Twistor geometry of a pair of second order ODEs, Comm.
Math. Phys. 321, (2013) 681–701. arXiv:1203.4158.
[9] E. Cartan, Sur une classe d’espaces de Weyl, Ann. Sci. E´cole Norm. Sup. (3) 60 (1943) 1–16.
[10] E. Cartan, La geometria de las ecuaciones diferenciales de tercer orden, Revista Mat. Hisp.-Amer.
4 (1941) 3-33.
[11] E. Cartan, Les syste`mes diffe´rentiels exte´rieurs et leurs applications ge´ome´triques, Actualite´s, Sci.
Ind., (Paris: Hermann) no. 994 (1945).
[12] A. Chudecki and M. Przanowski, Killing symmetries in H-spaces with Λ, J. Math. Phys. 54, 102503
(2013); doi: 10.1063/1.4826346.
[13] D.M. DeTurck and D. Yang, Existence of elastic deformations with prescribed principal strains and
triply orthogonal systems, Duke Math. J. 51, no. 2 (1984) 243–260.
[14] B. Doubrov and E.V. Ferapontov, On the integrability of symplectic Monge-Ampe`re equations, J.
Geom. Phys. 60 (2010) 1604-1616.
[15] M. Dunajski, The twisted photon associated to hyper-Hermitian four-manifolds, J. Geom. Phys. 30,
no. 3 (1999) 266–281.
[16] M. Dunajski, An interpolating dispersionless integrable system, J. Phys. A 4, no. 31 (2008) 315202.
13
[17] M. Dunajski, L.J. Mason and P. Tod, Einstein-Weyl geometry, the dKP equation and twistor theory,
J. Geom. Phys. 37, no. 1–2 (2001) 63–93.
[18] M. Dunajski, A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic
type, J. Geom. Phys. 51, no. 1 (2004) 126-137.
[19] M. Dunajski, Anti-self-dual four-manifolds with a parallel real spinor, Proc. Roy. Soc. Lond. A 458
(2002) 1205–1222.
[20] E.V. Ferapontov and K.R. Khusnutdinova, On the integrability of (2+1)-dimensional quasilinear
systems, Comm. Math. Phys. 248 (2004) 187–206.
[21] E. V. Ferapontov, V. Novikov and I. Roustemoglou, On the classification of discrete Hirota-type
equations in 3D, arXiv:1312.1574 (2013).
[22] E. V. Ferapontov and B. Kruglikov, Dispersionless integrable systems in 3D and Einstein-Weyl
geometry, to appear in J. Diff. Geom. (2014); arXiv:1208.2728v3.
[23] D. Finley and J. F. Plebanski, Further heavenly metrics and their symmetries J. Math. Phys. 17
(1976), 585.
[24] J.D.E. Grant and J.A. Vickers, Block diagonalization of four-dimensional metrics, Classical Quan-
tum Gravity 26, no. 23 (2009) 235014, 23 pp.
[25] D.A. Grossman, Torsion-free path geometries and integrable second order ODE systems, Sel. Math.,
New ser. 6 (2000) 399–442.
[26] N.J. Hitchin, Complex manifolds and Einstein’s equations, Twistor geometry and nonlinear systems
(Primorsko, 1980), 73–99, Lecture Notes in Math. 970, Springer, Berlin-New York (1982).
[27] M. Hoegner, Quaternion-Kaehler four-manifolds and Przanowski’s function, J. Math. Phys. 53
(2012) 103517.
[28] B. S. Kruglikov, V. V. Lychagin, Mayer brackets and solvability of PDEs – II , Trans. Amer. Math.
Soc. 358, no.3 (2005), 1077–1103.
[29] B. S. Kruglikov, V.V. Lychagin, Dimension of the solutions space of PDEs , ArXive e-print:
math.DG/0610789; In: Global Integrability of Field Theories, Proc. of GIFT-2006, Ed. J.Calmet,
W.Seiler, R.Tucker (2006), 5–25.
[30] W. Krynski, Paraconformal structures and differential equations, Diff. Geom. Appl. 28 (2010),
523531.
[31] S.V. Manakov and P.M. Santini, The Cauchy problem on the plane for the dispersionless Kadomtsev-
Petviashvili equation, JETP Lett. 83 (2006) 462–6.
[32] P. Nurowski, Differential equations and conformal structures, J. Geom. Phys. 55, no. 1 (2005) 19–49.
[33] M.V. Pavlov, J.H. Chang, Y.T. Chen, Integrability of the Manakov–Santini hierarchy,
arXiv:0910.2400.
[34] R. Penrose, Nonlinear gravitons and curved twistor theory, General Relativity and Gravitation 7,
no. 1 (1976) 31–52.
[35] J. F. Pleban´ski, Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975) 2395–2402.
14
[36] J.F. Pleban´ski and I. Robinson, Left-Degenerate Vacuum Metrics, Phys. Rev. Lett. 37 (1976) 493–
495.
[37] M. Przanowski, Locally Hermite Einstein, selfdual gravitational instantons, Acta Phys. Polon. B14
(1983) 625627.
[38] K.P. Tod, Einstein-Weyl spaces and third-order differential equations, J. Math. Phys. 41 (2000)
5572–5581.
[39] R.S. Ward, Einstein-Weyl spaces and SU(∞) Toda fields, Class. Quantum Grav. 7, no. 4 (1990)
L95-L98.
[40] R.S. Ward, Integrable and solvable systems, and relations among them, Philos. Trans. Roy. Soc.
London Ser. A 315, no. 1533 (1985) 451–457.
[41] G. Yi, P. M. Santini, The Inverse Spectral Transform for the Dunajski hierarchy and some of its
reductions, I: Cauchy problem and longtime behavior of solutions. J. Phys. A: Math. Theor. 48
(2015) 215203.
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