Flat $(2,3,5)$-Distributions and Chazy's Equations

In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or $(2,3,5)$-distributions determined by a single function of the form $F(q)$, the vanishing condition for the curvature invariant is given by a 6$^{\rm th}$ order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7$^{\rm th}$ order nonlinear ODE described in Dunajski and Sokolov. We show that the 6$^{\rm th}$ order ODE can be reduced to a 3$^{\rm rd}$ order nonlinear ODE that is a generalised Chazy equation. The 7$^{\rm th}$ order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat $(2,3,5)$-distributions not of the form $F(q)=q^m$. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split $G_2$ as their group of symmetries.


Introduction
The following 6 th order nonlinear ODE 10F (6) arises in [6,Corollary 2.1] in the study of generic 2-plane fields on 5-manifolds. The genericity condition here means F (q) = 0 in (1.1). This ODE arises as the integrability condition for generic 2-plane fields on 5 manifolds determined by a function of a single variable of the form F (q). A generic 2-plane field D on a 5-manifold M is a maximally non-integrable rank 2 distribution. For further details, see [6,18,24,25]. This determines a filtration of the tangent bundle given by The distribution [D, D] has rank 3 while the full tangent space T M has rank 5, hence such a geometry is also known as a (2, 3, 5)-distribution. Let M xyzpq denote the 5-dimensional manifold with local coordinates given by (x, y, z, p, q). The generic 2-plane field or rank 2 distribution determined by a function F (q) of a single variable with F (q) = 0 is given by D = span{∂ q , ∂ x + p∂ y + q∂ p + F (q)∂ z }.
The fundamental Cartan curvature invariant of this distribution is computed in [6] and is found to be the term in the left hand side of (1.1). It is known that equation (1.1) vanishes when in the context of investigating its Painlevé property. Solutions to equation (2.1) turn out to depend on hypergeometric functions. For further details, see [2] or [14]. Treat x as a dependent variable of s so that where z 1 (s), z 2 (s) are linearly independent solutions to the second order hypergeometric differential equation Here a, b, c are constants to be determined. The general solution to this ODE (2.2) is given by hypergeometric functions z(s) = µ 2 F 1 (a, b; c; s) + ν 2 F 1 (a − c + 1, b − c + 1; 2 − c; s)s 1−c .
Here µ, ν are constants. A computation gives where dot denotes derivative with respect to s. We deduce that Applying the derivative to Chazy's solution for y given by we find that (2.1) is satisfied precisely when (a, b, c) is one of  where P (a 1 ,b 1 ) n is the Jacobi polynomial. Taking z 1 (s) = ν 2 F 1 (1 − c, a + 1 − c; 2 − c; s)s 1−c and z 2 (s) = µ, a computation shows that Switching back to the original independent variable x, this gives as one solution to (2.1). This solution is invariant under translations of the form x → x + C.
In [12,13], Chazy also observed that is a solution to (2.1). It is well-known that Chazy's equation and its generalised version can be rewritten as a first order system. This provides different parametrisations of y, in addition to the solution (2.3) originally given by Chazy. This will be discussed in Section 5. The method discussed here can also be applied to the generalised Chazy equation.

Generalised Chazy equations
The generalised Chazy equation is given by for k = ±6. We have the following: where z 1 (s), z 2 (s) are linearly independent solutions to the hypergeometric differential equation Then Proof . Analogous to solving Chazy's equation (2.1), we find that the generalised equation (3.1) holds provided For a, b = 0, solving the system of equations 6((a − b)k − 6(a + b))((a − b)k + 6(a + b)) = 0, (24ab − 12(a + b)c + 5(a + b) + (2c − 1))k 2 + 432(a + b)c − 180(a + b) − 72c + 36 = 0, gives the list of (a, b, c) as above. We exclude the case where (a, b, c) = 0, 0, 1 2 . Note that interchanging a and b gives the same solution so that the full list is symmetric in a and b.
When either a or b is zero, we again get y(x) = − 6 x as a solution. In [12,13], Chazy noted that is also a solution to (3.1). As a corollary to Proposition 3.1, we have Then A Painlevé type analysis of equation (3.1) as done in [14] shows that the leading orders for analytic solutions to (3.1) occur at −6, −3 + k 2 or −3 − k 2 . This corresponds to solutions of (3.1) given by x .
These solutions are invariant under translations of the form x → x + C. In the case of k = ± 2 3 obtained in (1.2), we have Along with the zero solution I(q) = 0, these solutions correspond respectively (modulo constants of integration) to the well-known explicit solutions to (1.1): For these functions of a single variable q the associated (2, 3, 5)-distributions have vanishing Cartan invariant and therefore have G 2 as their local symmetry.

Relationship to ODE studied by Dunajski and Sokolov
For the function y = y(t), the 7 th order nonlinear ODE studied in [16] is given by This is the unique 7 th order ODE admitting the submaximal contact symmetry group of dimension ten (see [16,21]) and its relationship to equation (1.1) was originally explored in [6]. It is instructive to consider the 6 th order ODE (for the Legendre transformation later on): 10 with H(t) = y (t). Let us show that this ODE can be reduced to a generalised Chazy equation. Again working locally in an open set where y (t) is non-zero, and assuming y (t) to be positive, we can make the substitution e p(t) = y (3) to get We note that this 4 th order ODE historically appears in [9, Section XII, formula (12)], where it first arises as the obstruction to integrability for (2, 3, 5)-distributions of the form D F (q) . This will be made clear below once we show that (4.2) is the Legendre transform of (1.1) [6] and we will discuss this further in Section 6. Thus, for v(t) = p (t), we obtain the third order ODE Rescaling v(t) by u(t) = 3 2 v(t), we put (4.4) into the normal form We therefore see that the ODE that Dunajski and Sokolov study in [16] reduces to a generalised Chazy equation (4.5) with parameter k = ± 3 2 , related to the generalised Chazy equation (1.2) just by taking the reciprocals (k ) 2 = 1 k 2 of the corresponding parameters. Let t(s) = w 2 (s) w 1 (s) where w 1 (s), w 2 (s) are linearly independent solutions to the hypergeometric differential equation The solution to (4.5) is then given by u = 6 d dt log w 1 . A similar leading order analysis as before shows that the leading orders occur at This corresponds to solutions of (4.5) given by Along with the zero solution u(t) = 0, these correspond respectively (modulo constants of integration) to solutions of (4.2) given by In [6, Proposition 2.2], it is shown that a Legendre transformation takes (1.1) to (4.1). Hence we may hypothesise that amongst all 3 rd order generalised Chazy equations, only those with the parameters k = ± 3 2 , k = ± 2 3 have in addition solutions that can be obtained from the dual equation via a Legendre transform. Proof . Applying the exterior derivative to the relation gives (F − t)dq + (H − q)dt = 0, so that we take F = t, H = q and applying 3 , etc. A computation shows that the 6 th order ODE (1.1) holds for F iff (4.2) holds for H.
In light of the solutions obtained by solving the generalised Chazy equations, we can pass to where q = z 2 (s) z 1 (s) and I(q) = 6 d dq log z 1 are given in Corollary 3.2. This gives Similarly, for the dual equation (4.2) under the Legendre transform we pass to where t = w 2 (s) w 1 (s) and u(t) = 6 d dt log w 1 are solutions to (4.5). This gives We have There exists a Legendre transformation between Chazy's solutions of (1.2) and (4.5) given by taking This defines a mapping solves the 6 th order ODE (4.2). For the converse, the Legendre transform is given by In particular, if u(t) = 6 d dt log w 1 solves the dual ODE (4.5), then I(q) = 6 d dq log z 1 solves (1.2). Hence, if H(t) = (w 1 ) 4 dtdt solves the 6 th order ODE (4.2), then Proof . We observe that as a consequence of Chazy's solutions, the Legendre transform in Proposition 4.1 gives and thereforė Together this yields (z 1 ) 3 = (w 1 ) −4 , from which we deduce For the converse, we find The rest follows from a routine computation.
In [16, formula (8)], a family of solutions to (4.1) is found to be given by the algebraic curve with a = b, and f (t) a quadratic. This gives We obtain a solution to (4.1) with We find that for this solution, it yields as a solution to the generalized Chazy's equation with parameter k 2 = 9 4 . This corresponds to the solution given by Chazy in (3.2). It will be interesting to determine the solutions of (4.5) from the general solution given by [16, formula (13)].

First order system and dif ferent parametrisations of Chazy's equations
In this section, we first show that the generalised Chazy equation is equivalent to solving a third order differential equation involving the Schwarzian derivative and a potential term V (s). It is well-known that solutions to the generalised Chazy equation (3.1) can be rewritten as a first order system. For further details, see [2]. The first order system provides different parametrisations of the solutions, in addition to the one given by (2.3). We compute the solutions to the generalised Chazy equation (1.2) with k = ± 2 3 for the different parametrisations below and present them in Tables 1, 2 and 3. We also show how these solutions are related to one another by algebraic transformation of hypergeometric functions.
Let Ω 1 , Ω 2 , Ω 3 be functions of q. Let dot denote differentiation with respect to q. Then consideṙ and α, β, γ are constants. Introducing the parameter we find that , The system of equations (5.1) are satisfied iff is the Schwarzian derivative of s(q) and Switching independent and dependent variables in (5.2), we have so that the dual of (5.2) is The general solution is given by where u 1 , u 2 are linearly independent solutions of the second order ODE The general solution of (5.3) suggests taking In [2], it was determined that taking gives solutions to the generalised Chazy equation with only the first coinciding with the list in Proposition 3.1. This suggests that the solutions to (5.2) are more general than the solution of the form y = 6 d dq log z 1 given by Chazy [13]. We can express Chazy's solution in terms of s(q) as follows. A computation of the Wronskian of linearly independent solutions z 1 , z 2 to (2.2) gives for some non-zero constant w 0 . The latter equality holds by solving the first order differential equation See for instance [4]. From q(s) = z 2 (s) Applying this derivative to s(q), we obtain . Hence and we geẗ Therefore we have A comparison of Chazy's formula for y = 6 d dq log z 1 with the formula for y in (5.4) suggests taking c = 2 3 , a + b = 1 3 . This is satisfied by (a, b, c) = k−6 6k , k+6 6k , 2 3 in Proposition 3.1. For k = 2 3 as in (1.2), we get the following solutions for u given in Table 1. For this Table 1. Chazy parameter, the hypergometric series truncate and the solutions to (5.3) can be given by elementary functions. We have Moreover, the solutions in each row are related to one another by algebraic transformations of hypergeometric functions. The solutions in the second, third and fourth rows of Table 1 can be obtained from the first by a cubic transformation of hypergeometric functions (see [23, formula (23)]). Explicitly, to show how the solution in the third row is related to the first, let ω be a cube root of unity (solution to ω 2 + ω + 1 = 0) and consider the map A different parametrisation of Chazy's equations (cf. [10]) is also given by A comparison with Chazy's formula (2.3) yields (a, b, c) = k−6 12k , k+6 12k , 2 3 from Proposition 3.1. The solution of the form (5.5) solves the generalised Chazy equation (3.1) whenever (α, β, γ) in (5.2) is given by The solution to (5.2) with (α, β, γ) = 1 k , 1 3 , 1 2 is given by the Schwarz function J (see [2,3]). Considering the symmetry s = 1 − K brings (5.5) to and comparing with Chazy's formula gives (a, b, c) = k−6 12k , k+6 12k , 1 2 from Proposition 3.1. The solution of the form (5.6) solves (3.1) whenever (α, β, γ) in (5.2) is given by The symmetry s = 1 − K permutes β and γ in the formula for V (s) in (5.2). For k = 2 3 , the solutions to (5.3) with the parametrisation by (5.5) are presented in Table 2.  We also note that we have The algebraic transformations relating the solutions between the rows are given as follows. The map that takes the solution from the second row to the solution in the first row of Table 2 is a composition of fractional linear transformations and cubic transformation due to Goursat (see [23, formulas (20)

Thenz(t) satisfies (2.2) with (a, b, c)
To obtain the solution given in the third row from those in the first row requires a transformation of degree 4 (see [   to the solution with (a, b, c) = − 1 3 , − 2 3 , 1 2 and an Euler transformation followed by s → 1 − s again takes this to the solution with (a, b, c) = 5 6 , − 2 3 , 2 3 as given in row 1 of Table 2. Finally, consider the parametrisation given by We find that (α, β, γ) is one of , we obtain (a, b, c) = (−4, 2, − 1 2 ). Let us relate the solution to the differential equation (2.2) with (a, b, c) = (−4, 2, − 1 2 ) to the solution given in the second row of Table 2. For (α, β, γ) and thus for k = 2 3 , we have  Table 2 given by the map and the relatioñ We Let us summarise the solutions to (5.3) with parametrisation given by (5.7) in Table 3.
We can consider conformal rescalings of the metric such thatĝ D F (q) = Ω 2 g D F (q) is Ricci flat. It turns out that if we take Ω = ν(q) −1 > 0, then the Ricci tensor of the rescaled metricĝ D F (q) is given by so the appropriate conformal scale ν(q) can be found by solving the differential equation in (6.6) (cf. [24,Proposition 35]). In the first part of this section we consider the conformally flat metrics (6.5) obtained by solving (1.2) using the solution (2.3). Next, we then consider the solutions obtained from different parametrisations of the generalised Chazy equation given by (5.4), (5.5) and (5.7). We also consider conformally flat metrics obtained from solving the Legendre transform of (1.2). This involves computing the coframe for the metric under the Legendre transform. Finally, we consider the metrics obtain from Chazy's solutions given by (3.2). The metrics associated to (2, 3, 5)-distributions D F (q) of the form F (q) = q m where m ∈ −1, 1 3 , 2 3 , 2 are given in [18].

Chazy's solution
In order to express Nurowski's metric associated to flat (2, 3, 5)-distributions obtained from solving (1.2), we have to switch independent variable s and dependent variable q. In other words we pass to coordinates (x, y, z, p, s) with q(s) = z 2 (s) z 1 (s) where z 1 (s), z 2 (s) are given in Corollary 3.2. For this parametrisation we have Let us denote Supplement by the 1-forms Then Nurowski's metric (6.5) has vanishing Weyl tensor (and hence conformally flat) and D C has the split real form of G 2 as its group of local symmetries.
Let us provide an explicit example given by Corollary 3.2. It turns out for the values of (a, b, c) obtained in Corollary 3.2, the solutions can be given by elementary functions. For (a, b, c) = − 2 3 , 5 6 , 1 2 , the solutions to the hypergeometric differential equation (2.2) are given by where P m and Q m are the associated Legendre functions. This suggest passing further to the variable r = √ s, in which case (2.2) with (a, b, c) = − 2 3 , 5 6 , 1 2 becomes 1 4 ( The general solution is now given by the elementary functions Then Nurowski's metric (6.5) has vanishing Weyl tensor (and hence conformally flat) and D 0 has the split real form of G 2 as its group of local symmetries. The Ricci tensor for this metric is Rescaling this metric by where a 1 and a 2 are constants, the conformally rescaled metricĝ D 0 = Ω 2 g D 0 given bŷ is both Ricci-flat and conformally flat.
Then Nurowski's metric has vanishing Weyl tensor (and hence conformally flat) and D s has the split real form of G 2 as its group of local symmetries.
To obtain explicit examples, it is useful to switch the independent variable q and the dependent variable s. We pass to the variables (x, y, z, p, s) with q = u 2 (s) u 1 (s) where u 1 and u 2 are linearly independent solutions of (5.3) given in Table 1. Note that up to fractional linear transformations in the variable s, we only need to consider the solutions to (5.2) with the values of (α, β, γ) given by either (3, 3, 3) or 3, 1 3 , 1 3 for the parametrisation given by (5.4). Note the symmetry permuting β and γ.
A computation shows that W (u 1 , u 2 ) = u 1u2 −u 1 u 2 is constant, which we can normalise to set W (u 1 , u 2 ) = 1. We have Let us denote ds.
Theorem 6.4. Let u 1 (s), u 2 (s) be two linearly independent solutions to (5.3) subject to the constraint W (u 1 , u 2 ) = 1 with (α, β, γ) given by Table 1. Let D s denote the (2, 3, 5)-distribution on M xyzps associated to the annihilator of We pass to the annihilator 1-forms given by (6.4) to obtaiñ Take the coframe on M xyzps to be given by Then Nurowski's metric has vanishing Weyl tensor (and hence conformally flat) and D s has the split real form of G 2 as its group of local symmetries.
Supplement by the 1-forms and take the coframe on M xyzpq to be given by (θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ) as in (6.2). Then Nurowski's metric (6.3) has vanishing Weyl tensor (and hence conformally flat) and D s 1 has the split real form of G 2 as its group of local symmetries.
Similarly, for I(q) given by (5.7), that is to say I(q) = d dq logṡ

An-Nurowski circle twistor bundle
In [5], An and Nurowski showed how to associate to a split signature conformal structure [g] on a 4-manifold M 4 a natural (2, 3, 5)-distribution. 4-dimensional split signature conformal structures admit real self-dual totally null 2-planes. The bundle of such 2-planes is a circle bundle over M 4 with fibres S 1 [5]. This is called the circle twistor bundle T(M 4 ) and it has a rank 2 distribution given by lifting horizontally the null 2-planes on M 4 . This distribution is non-integrable, i.e., defines a (2, 3, 5)-distribution whenever the self-dual part of the Weyl tensor of g on M 4 is non-vanishing. Moreover in [6], the authors presented split signature conformal structures on M 4 that give rise to (2, 3, 5)-distributions of the form D F (q) on T(M 4 ). Such split signature metrics are called Plebański's second heavenly metrics in [6]. Following [6, Section 3], we can find these metrics that have a flat circle twistor bundle. Such circle twistor bundles have split G 2 as their group of symmetries. Let (w, x, y, z) be local coordinates on M 4 . Let Θ = Θ(w, x, y, z) be an arbitrary function of 4 variables (second heavenly function of Plebański). Let (e i ) be an orthonormal frame on M 4 and (θ j ) the dual coframe satisfying θ j (e i ) = δ j i . The split signature Plebański metric is given by where θ i θ j = 1 2 θ i ⊗ θ j + 1 2 θ j ⊗ θ i and θ 1 = dx − Θ yy dw + Θ xy dz, θ 2 = dw, θ 3 = dy − Θ xx dz + Θ xy dw, θ 4 = dz.