The Chazy XII Equation and Schwarz Triangle Functions

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Introduction
In 1911, J. Chazy [10] considered the classification problem of all third order differential equations of the form y = F (z, y, y , y ) possessing the Painlevé property, where the prime denotes d/dz, and F is a polynomial in y, y , y and locally analytic in z. A differential equation in the complex plane is said to have the Painlevé property if its general solution has no movable branch points. In his work, Chazy introduced thirteen classes of equations referred to as Chazy classes I-XIII. Among these, classes III and XII are particularly interesting as their general solutions possess a movable barrier, i.e., a closed curve in the complex plane across which the solutions can not be analytically continued. That is, the solutions are analytic (or meromorphic) on either side of the barrier depending on the prescribed initial conditions. Both the Chazy III and XII equations can be expressed together as y − 2yy + 3y 2 = K 6y − y 2 2 , K = 0 or K = 4 36 − k 2 . (1.1) for k ∈ C ∪ ∞, k = 0. He also showed that (1.1) possesses the Painlevé property when K = 0 or when the value of the parameter k is a positive integer such that k > 1 and k = 6 (see also [12]). The k = 0 case is linearizable by Airy's equation χ = csχ, c constant [11], and the limiting cases of k = ±6 can be solved via elliptic functions [10,12]. For K = 0 or for integer values k > 6, the general solution of (1.1) possesses a movable natural barrier in the complex plane. The Chazy III equation arises in mathematical physics in studies concerning magnetic monopoles [3], self-dual Yang-Mills and Einstein equations [8,21], topological field theory [14], as well as special reductions of hydrodynamic type equations [16] and incompressible fluids [30]. The Chazy XII equation is related to the generalized Darboux-Halphen system [1,2], which arises in reductions of self-dual Yang-Mills equations associated with the gauge group of diffeomorphism Diff(S 3 ) of a 3-sphere [9] as well as SU(2)-invariant hypercomplex manifolds [22] with self-dual Weyl curvature [6]. More recently, the Chazy XII equation with specific values of the parameter k has been linked to the study of vanishing Cartan curvature invariant for certain types of rank 2 distributions on a 5-manifold [28,29]. Consequently, there has been renewed interest in the study of (1.1) and the singularity structure of its solutions in the complex plane. Interested readers are referred to the comprehensive works by Bureau [4], Clarkson and Olver [11] and Cosgrove [12].
A significant aspect of (1.1) is the fact that its general solutions can be expressed in terms of Schwarz triangle functions which define conformal mappings from the upper half complex plane onto a region bounded by three circular arcs (see, e.g., [24]). For K = 0, solutions of the Chazy III equation are related to the automorphic forms associated with the modular group SL 2 (Z) and its subgroups [7]. This fact can be traced back to the work of S. Ramanujan. In 1916, Ramanujan [25], [27, pp. 136-162] introduced certain functions P (q), Q(q), R(q), q := e 2πiz , Im(z) > 0, which correspond to the (first three) Eisenstein series associated with the modular group SL 2 (Z). He showed that these functions satisfy the differential relations System (1.2) can be reduced to a single third order differential equation for P (q). In fact, the function y(z) := πiP (q) satisfies the Chazy III equation. For K = 0, the solutions of the Chazy XII equation are also related to Schwarz functions automorphic on curvilinear triangles that tessellate the interior of the natural barrier for integer values of the parameter k > 6. For 2 ≤ k ≤ 5 the solutions of (1.1) are expressed via the polyhedral functions which are rational functions associated with symmetry groups of solids (see, e.g., [17]).
In this paper, we primarily consider the Chazy XII equation (1.1) for integer values k > 6. The main objective of this paper is to find all solutions y(z) that are given in terms of Schwarz triangle functions. The central result to achieve this goal is the following: If y(z) = 6Y (z) solves (1.1) with K = 0, then Y (z) is a convex linear combination where s(z) = S(α, β, γ; z) is the Schwarz triangle function (see Section 2.2 for details). Thus, Table 2 provides explicit parametrizations of the Chazy XII solutions in terms of a distinct family of Schwarz triangle functions. This family includes previously known functions, e.g., S( 1 2 , 1 3 , 1 k ; z), 6 < k ∈ N as well as some new cases. We believe that our approach is new, and that it can be applied to study similar nonlinear equations with natural barrier [5,23].
The paper is organized as follows. Section 2.1 develops necessary geometric background on affine connections on a one-dimensional complex manifold with a projective structure following [14]. In Section 2.2, the generalized Darboux-Halphen system is introduced and its solutions in terms of Schwarz triangle functions are discussed. The connection between the Chazy XII equation and the generalized Darboux-Halphen system described in Section 3.1 leads to a purely algebraic formulation of (1.1). The solvability conditions of this algebraic system are analyzed in Section 3.2, and these lead to a classification of the Chazy XII solutions according to their pole structures inside the natural barrier. In Section 3.3, a Ramanujan-type triple of functions is introduced. These functions satisfy a system of first order equations that is similar to (1.2) and is equivalent to the Chazy XII equation. In Section 4.1, the Schwarz triangle function is presented as the inverse to the conformal map defined by the ratio of two linearly independent solutions of the hypergeometric equation. The parametrizations of the Ramanujan-like triple as well as the Chazy XII solution in terms of hypergeometric functions are also discussed here. Section 4.2 outlines the rational transformations between the distinct Schwarz functions parameterizing the Chazy XII solution. These rational maps are derived from the well-known algebraic transformations among the hypergeometric functions. In order to make the paper self-contained, we have included two appendices. Appendix A contains a proof of Lemma 3.1 introduced in Section 3.2, while Appendix B contains an elementary derivation of the radius of the natural barrier for the Chazy XII solution.

Af f ine connection and projective structure
We begin this section by reviewing the relation between the solution y(z) in (1.1) with affine connections on a one-dimensional complex manifold. We consider differential forms of order m ∈ N denoted by f = f (z)dz m on a one-dimensional complex manifold with local coordinate z, where f (z) is a holomorphic (or meromorphic) function. Under the local change of coordinates z →z(z), f transforms according to f (z)dz m =f (z)dz m . The covariant derivative of a mdifferential is a (m + 1)-differential ∇f defined by where η = η(z)dz is a holomorphic (or meromorphic) affine connection on the manifold. It follows from the transformation property of ∇f that η must transform as The "curvature" associated with η is defined by the quadratic differential Ω = Ω(z)dz 2 as In fact, Ω is a projective connection transforming under local change of coordinates as for any constant J. This yields the following third order differential equation for η(z): which reduces to (1.1) by setting y = 3η and 108K = (12 − J). Thus, the Chazy III and Chazy XII equations (1.1) can be interpreted as a certain differential polynomial invariant of an affine connection in a one-dimensional complex manifold with a projective structure. A projective structure on a one-dimensional complex manifold M ⊆ CP 1 is defined by an atlas of local coordinates with transition functions given by Möbius transformations. In a local coordinate chart, the Möbius transformations are generated by the vector fields ∂ z , z∂ z , z 2 ∂ z isomorphic to the Lie algebra sl 2 (C). A nontrivial representation of sl 2 (C) is given by the vector fields u 2 1 ∂ s , u 1 u 2 ∂ s , u 2 2 ∂ s where u 1 (s), u 2 (s) is a pair of linearly independent solutions of the complex Schrödinger equation (where the factor 1 4 is inserted for convenience). Then the identification of the vector field ∂ z with u 2 1 ∂ s induces a local change of coordinates s → z(s) on M via the ratio If M is not simply connected, the monodromy group G ⊂ GL 2 (C) resulting from the analytic extensions of the pair (u 2 , u 1 ) → (au 2 + bu 1 , cu 2 + du 1 ) along all possible closed loops in M acts projectively on the ratio in (2.6) via the Möbius transformations The projectivized monodromy group is the quotient group Γ ∼ = G/λI 2 ⊆ PSL 2 (C), where I 2 is the 2 × 2 identity matrix. A different choice for the basis (u 1 , u 2 ) would lead to a different projective structure on M . Note that the Schwarzian derivative is a differential invariant of the projective transformation, i.e., {γ(z); s} = {z; s}. For a projective structure induced by the Schrödinger equation (2.5), z(s) satisfies the third order Schwarzian equation which can be linearized via (2.5) and (2.6). It follows that the inverse function s(z) (if it exists globally) is a projective invariant function on the manifold M with the automorphism s(z) = s(γ(z)), γ ∈ Γ, and satisfies An affine connection on the one-dimensional complex manifold M can be defined uniquely by its trivialization η(s) = 0 in the projective invariant coordinate s. Then the transformation s → z(s) = u 2 (s)/u 1 (s) in terms of the solutions of the Schrödinger equation leads to the following expression of the affine connection in the z-coordinate η(z) = s (z)/s (z) = −z (s)/z (s) 2 = 2u 1 (s)u 1 (s), using the transformation rule (2.1). Furthermore, from (2.2), the differential Ω(z) is then given in terms of the Schrödinger potential V (s) as Consequently, the solutions of the Chazy III and XII equations (1.1), which are equivalent to (2.4), can be expressed via the solutions and the potential of the complex Schrödinger equation (2.5). It is worth noting that if the affine connection is trivialized in a different coordinate x(s) with dx = f (s)ds, then from (2.1), η(z)dz = d log(f (s)s (z)). We will utilize this fact and the geometric framework discussed above to construct a solution method in Section 3 for (1.1), by exploiting its relationship to another nonlinear differential system described below.
The gDH system can be solved via the Schrödinger equation (2.5) with the potential and defining a projective structure on M via the ratio z(s) in (2.6) as described in Section 2.1. Note that in this case (2.5) is a second order Fuchsian differential equation with three regular singular points, and α, β, γ are the exponent differences (for any pair of linearly independent solutions u 1 and u 2 ) prescribed at the singular points 0, 1 and ∞, respectively. The generators of the projectivized monodromy group Γ are determined by the exponent differences. If the gDH variables w 1 (z), w 2 (z), w 3 (z) are expressed in terms of the projective invariant inverse function s(z) (and its derivatives) as follows: , then a straightforward calculation shows that (2.9) reduces to the Schwarzian equation (2.8b) for s(z), where the constants α, β, γ in V (s) are the same as those appearing in τ 2 of (2.9). Equation (2.8b) is equivalent to (2.8a) (after interchanging the dependent and independent variables) which is then reduced to (2.5). Thus, the gDH system can be effectively linearized by the Schrödinger equation (2.5) with the potential V (s) given by (2.10). The ratio z(s) in (2.6) of any two linearly independent solutions u 1 , u 2 of (2.5) with the potential V (s) given by (2.10) defines a conformal mapping that was studied extensively by H.A. Schwarz [31] in 1873. The map z(s) is, in general, branched at the regular singular points s = 0, 1, ∞. However, if the parameters α, β, γ are either zero or reciprocals of positive integers, and satisfy α + β + γ < 1, then the mapping z(s) defines a plane region D, which is tessellated by an infinite number of non-overlapping hyperbolic, circular triangles on the complex z-plane. The interior of each triangle is an image of the upper (lower)-half s-plane under the map z(s) and its analytic extensions, and is bounded by three circular arcs forming interior angles απ, βπ, and γπ at the vertices z(0), z(1), and z(∞), respectively. Two adjacent triangles are obtained via Schwarz reflection principle, and are images of each other under reflection across the circular arc that forms their common boundary. In this case, the inverse s(z) is a single-valued, meromorphic, automorphic function whose automorphism group is the projective monodromy group Γ associated with (2.5) and (2.10). That is, s(γ(z)) = s(z) for all γ ∈ Γ where γ(z) is the Möbius transformation defined in (2.7). When the exponent differences satisfy the conditions prescribed above, Γ is a discrete subgroup of PSL(2, R), and turns out to be the group of Möbius transformations generated by an even number of reflections across the boundaries of the circular triangles. This automorphic inverse function is called the Schwarz triangle function, and the automorphism group Γ is referred to as the triangle group. It is worth noting that if α, β, γ in (2.10) are either zero or reciprocals of positive integers, but either α + β + γ = 1 or α + β + γ > 1, then the map z(s) in (2.6) tiles the z-plane either into infinitely many plane triangles, or the extended z-plane (Riemann sphere) into a finite number of spherical triangles, respectively. The triangle group Γ is a simply or doubly periodic group when α + β + γ = 1, and corresponds to one of the four symmetry groups of the regular solids when α + β + γ > 1. A detailed discussion of the automorphic groups can be found in the monograph [17].
It follows from (2.8b) or from (2.5) with V (s) as in (2.10) that the only possible singularities of s(z) and its derivatives on the domain D are located at the vertices of each triangle where s(z) takes the value of 0, 1, or ∞. The boundary of D in the z-plane is a Γ-invariant circle C which is orthogonal to all three sides of each triangle and its reflected images. This orthogonal circle C is the set of limit points for the automorphic group Γ, and corresponds to a dense set of essential singularities which form a natural barrier for the function s(z). In its domain of existence D, the only possible singularities of s(z) are poles which correspond to the vertices where s(z) = ∞.
In summary, when the parameters α, β, γ in (2.10) are either zero or reciprocals of positive integers and α + β + γ < 1, the general solution of (2.8b) is obtained as the unique inverse of the ratio where u 1 and u 2 are two linearly independent solutions of (2.5). The solution is single-valued and meromorphic inside a disk in the extended z-plane, and can not be continued analytically across the boundary of the disk. This boundary is movable as its center and radius are completely determined by the initial conditions, which depend on the complex parameters a, b, c, d.
Recently, a number of new nonlinear differential equations whose solutions possess movable natural boundaries have been found [5,23]. These can be solved by first transforming them into a Schwarzian equation (2.8b) and then following the linearization scheme described above.

Chazy XII and triangle functions
In this section we present a solution method for the Chazy XII equation based on the geometric approach discussed in Section 2. Specifically, we exploit the fact that the Chazy XII equation in (1.1) is equivalent to the coordinate independent form in (2.3) for the quadratic differential Ω. We express the affine connection η associated with Ω via the gDH variables introduced in Section 2.2 so that the Chazy XII equation can be expressed in terms of the Schrödinger potential V (s) in (2.10) and its derivatives, in a simple algebraic fashion. Then the solution of the Chazy XII equation is obtained via the Schwarz triangle function S(α, β, γ; z) for specific values of the triple (α, β, γ), which we identify. Recall that in this article we only consider the Chazy XII equation. The parametrization of the Chazy III equation by Schwarz triangle functions was studied in [7].

The gDH system and the Chazy XII equation
Note first from (2.11) and (2.1) that 2w i = d log(f i (s)s (z))/dz transforms as an affine connection that is trivialized in the coordinate .
In particular, under a Möbius transformation z → γ(z) given by (2.7) with ad−bc = 1, w i transforms as which leaves the gDH system (2.9) invariant. Next let us define the function in terms of the gDH variables w i , where the coefficients a i are nonnegative constants. Then from (2.11), y(z) can be expressed in terms of s(z) and its derivatives as follows: with p := a 1 + a 2 + a 3 , α 1 := a 2 /p and β 1 := a 1 /p and 0 ≤ α 1 , β 1 ≤ 1. If we set p = 6 above, then it is possible to define an affine connection η such that in the z-coordinate, which are rational functions of s. The curvature in the z-coordinate is obtained using the transformation property (2.2) as where the second equality above follows from (2.8a) with V (s) as in (2.10). Then the covariant derivatives of the quadratic differential Ω are given by the transformation rule where V m+2 (s) are rational functions of s defined recursively for n ≥ 2 as Our next goal is to derive conditions under which Ω(z) will satisfy the projective-invariant equation (2.3), equivalently, the affine connection η(z) will satisfy (2.4). Then the function y(z) = 3η(z) defined in (3.2) with p = 6 will solve the Chazy XII equation in (1.1). Henceforth, the value p = 6 will be used throughout the rest of this article.
It follows from the expressions for Ω(z) and its covariant derivatives obtained above, that for m = 2, (2.3) implies a simple algebraic relation between rational functions V 2 (s) and V 4 (s), namely, that should hold for all s. This condition imposes certain restrictions on the parameters (α, β, γ) and (α 1 , β 1 ) appearing in the functions V 2 (s) and V 4 (s). It is worth pointing out here that (3.5) together with (3.1) lead to the central result advertised in Section 1.
In what follows, we will systematically determine the sets of parameters for which (3.5) holds. In particular, we will identify the values of the triple (α, β, γ) in the Schwarz triangle functions S(α, β, γ; z) which determine the solution of the Chazy XII equation y(z) via (3.2).

The Schwarz function parametrization
where the parameters (α 1 , β 1 , γ 1 ) are defined as follows: Then V 4 (s) is readily computed from the recurrence relation in (3.4). Upon substituting the expressions for V 2 (s) and V 4 (s) into (3.5) and rationalizing the resulting expression, one finds that (3.5) is satisfied if and only if for all values of s. The last identity is equivalent to the vanishing of the coefficients u i , i.e., System (3.8) represents a set of coupled, algebraic equations for the parameters (α, β, γ), (α 1 , β 1 ) and J subject to the conditions and J = 12. The case J = 0 corresponds to K = 1 9 or k = 0 in (1.1). The Chazy XII equation for this case can be linearized via Airy's equation as mentioned in Section 1. Hence, the J = 0 case is distinct from the cases considered here which correspond to Schrödinger potential V (s) in (2.10) with three regular singular points. Furthermore, the following lemma is proven in Appendix A. Henceforth, we take J = 0 through the remainder of the main text of this paper.
Finally, assume that γ 1 = 0. Since JD 2 = 12γ 2 1 D where D := A + B + C (see Lemma 3.2), it follows that D = 0. Then, (3.6) implies that γ = γ 1 = 0, contradicting Lemma 3.2. Lemma 3.2 implies that α, β, γ ∈ { 1 n , n ∈ N} in (3.9). Let s(z) = S(α, β, γ; z) be the triangle function associated with those parameters, and let z 0 be a vertex of a triangle in the domain of existence D of s(z) such that s(z 0 ) := s 0 ∈ {0, 1, ∞}. The map z(s) defined via the equations (2.6), (2.5) and (2.10), behaves locally near s = s 0 as (see, e.g., [24]) where ψ i is analytic near s 0 with ψ i (s 0 ) = 0, and µ ∈ {α, β, γ} is the corresponding exponent difference at the singular point s 0 . Consequently, the inverse s(z) is single-valued function defined locally as where φ i (z) is analytic in the neighborhood of z = z 0 with φ i (z 0 ) = 0, and m = µ −1 ∈ N. Thus, s − s 0 has a zero of order 1 α , 1 β at the vertices z 0 = z(0), z(1), respectively, and s(z) has a pole of order 1 γ at the vertex z 0 = z(∞) of each triangle in the domain D. As mentioned earlier in Section 2, it is sufficient to examine the behavior of the function s(z) and its derivatives near the vertices z(0), z(1), and z(∞) of just one triangle inside the domain D. The Schwarz reflection principle and the automorphic property then ensure that s(z) will have the same behavior at the vertices of the reflected triangles in D. A straightforward calculation using the above behavior of s(z) in (3.2) shows that y(z) is a meromorphic function in D with simple poles at the vertices z(0), z(1), z(∞) of each triangle in D with the residues Of course, at the boundary of the domain D, y(z) inherits the same natural barrier of essential singularities as the function s(z). Note that it is possible for y(z) to be analytic at a vertex z 0 in the interior of D provided that the residue vanishes at z 0 , i.e., if either α = α 1 , β = β 1 , or γ = γ 1 . However, it is impossible for all three residues to vanish simultaneously because then α + β + γ = α 1 + β 1 + γ 1 = 1, which violates the condition α + β + γ < 1 in (3.9). The discussion above concerning the singularity structure of y(z) in the interior of its domain of definition D can be summarized in the following proposition. Hence, there are three distinct cases resulting from Proposition 3.4, namely, y(z) has a simple pole at (1) only one vertex z 0 ∈ {z(0), z(1), z(∞)}, (2) only two of the three vertices, and (3) all three vertices. The admissible parameters that satisfy (3.8) can be determined by considering each case separately.
The parameter values listed in each row are modulo all possible permutations of the vertices, which also permutes the triples (α, β, γ) and (α 1 , β 1 , γ 1 ) accordingly, but K is the same for all permutations. Except for the Case 1(a) which was known to Chazy [10] and Cases 1(b) and 3(b) which were found in [2] using a different method, the remaining cases are new to the best of our knowledge. Note that in some cases, the second column of Table 1 contains a parameter other than the reciprocal of a positive integer. The corresponding Schwarz function is then not single-valued in its domain of definition but the Chazy XII solution y(z) given by (3.2) still remains single-valued. Such multi-valued Schwarz functions are related to a singlevalued Schwarz function via a rational transformation. For example, S( 2 3 , 1 k , 1 k ; z) in Case 2(c) is related to the single-valued function S( 1 2 , 1 3 , 1 k ; z) in Case 1(a) via a degree-2 rational transformation [24] In this case the associated triangle in the z-plane with interior angles { 2π 3 , π k , π k } is divided along the bisector of the angle 2π 3 into two triangles each with interior angles { π 2 , π 3 , π k }. We shall discuss such rational transformations among the Schwarz triangle functions in Section 4.2, which correspond to decomposition of a curvilinear triangle into two or more similar sub-triangles. Each of the cases marked with an asterisk corresponds to a single valued triangle function but the parameter K is different from that considered by Chazy. Each such case can be transformed to one in the immediately preceding row by a rescaling k → k m for m = 2, 3 or 4.
It is useful to note 1 that if one rescales the Chazy function by introducing Y (z) = y(z)/6 then (1.1) takes the form Then using (3.1) and (3.7), Y (z) can be expressed as a convex linear combination of the gDH variables as follows where the possible values of (α 1 , β 1 ) are listed in Table 1.

Chazy XII and 3 × 3 systems
We now make a couple of observations regarding the equivalence between the Chazy XII equation and systems of first order equations. The first is the relationship between the Chazy XII solution y(z) and the gDH variables w i given by (3.1). Note that for each case in Table 1, there is a gDH system (2.9) defined by the triple (α, β, γ). Then from (3.1) with coefficients given by a 1 = 6β 1 , a 2 = 6α 1 and a 3 = 6γ 1 , this gDH system can be reduced to the Chazy XII equation (1.1) with the corresponding Chazy parameter K listed in Table 1. It should be evident from Table 1 that there are several gDH systems corresponding to the same Chazy XII equation. This fact is illustrated in Table 2, where we list the combinations of the gDH variables w i which give the same y(z) satisfying the Chazy XII equation with parameter K = 4 36−k 2 . However, in each case the w i satisfy the gDH system (2.9) with a different function τ 2 parameterized by the triple (α, β, γ) listed in the corresponding row of Table 2. The last column of Table 2 lists the function φ(z) whose logarithmic derivative in (3.2) gives the solution y(z) for the Chazy XII equation. Since φ(z) are expressed in terms triangle functions S(α, β, γ; z) and their derivatives, it is possible for the same solution y(z) to be parametrized by two different triangle functions s 1 (z) and s 2 (z). For instance, Cases 1(a) and 1(b) correspond to s 1 (z) = S( 1 2 , 1 3 , 1 k ; z) and s 2 (z) = S( 1 3 , 1 3 , 2 k ; z). Then it follows from (3.2) that the corresponding φ(z) must be proportional. This leads to a differential relation between the two triangle functions, namely, , for some constant C, and can be solved to yield s 1 (z) = (2s 2 ( z) − 1) 2 , = C4 − 1 3 . In fact, all the triangle functions listed in Table 2 are related via rational transformations, which will be deduced in the next section by employing certain well-known transformations between solutions of hypergeometric equations.
Lastly, we introduce a Ramanujan-type system that is equivalent to the Chazy XII equation. Recall from Section 1 that the Ramanujan system (1.2) is equivalent to the Chazy III equation given by (1.1) with K = 0 if one identifies y(z) = πiP (q). It will be shown that both (1.2) and the differential system (3.12) introduced below have a simple geometric interpretation. Using the notations in Section 2.1, let us definê Then, if η(z) satisfies (2.4), the triple (P ,Q,R) satisfies the differential system (3.12) The first two equations in (3.12) are simply the definitions of the quadratic differential Ω = η − η 2 /2 and its covariant derivative ∇Ω = Ω − 2ηΩ associated with the affine connection η(z). The third one is equation (2.3) which is equivalent to the Chazy XII equation with J = 12−108K, K being the Chazy parameter. When J = 12, (3.12) reduces to the Ramanujan system (1.2) which is equivalent to the Chazy III equation. More explicitly, if y(z) is a solution of (1.1) then satisfy (3.12) which subsumes the Ramanujan system (1.2). The Ramanujan triple (P, Q, R) has a modular interpretation since it is related to the Eisenstein series associated with the modular group SL 2 (Z). The functionsP ,Q,R are also automorphic forms for the triangle group Γ and are parameterized by the triangle functions S(α, β, γ; z) via (3.2). In fact they transform as follows: P (γ(z)) = (cz + d) 2P (z) + 6c(cz + d) πi ,Q(γ(z)) = (cz + d) 4Q (z), However, we are not aware of any deep automorphic interpretation for (P ,Q,R) similar to that of the Ramanujan triple. Ramanujan [26] also gave a parametrization of his triple using the hypergeometric function 2 F 1 ( 1 2 , 1 2 , 1; s) that is related to the complete elliptic integral of the first kind. In the following section, we discuss the parametrizations of the Chazy XII solution y(z) as well as (P ,Q,R) in terms of hypergeometric functions.

Parameterization of Chazy XII solutions
Explicit solutions of the Chazy equation were presented in terms of the triangle functions listed in Table 1 of Section 3. Recall that the triangle functions satisfy the nonlinear third order equation given by (2.8b) with the potential V (s) as in (2.10). Equation (2.8b) is linearized via solutions of the Fuchsian equation (2.5) associated with V (s) using (2.12). Consequently, it is more convenient to express the Chazy solution y(s(z)) implicitly, that is, in terms of the variable s and a solution u(s) of the linear equation (2.5). Thus it is possible to treat the solutions of the nonlinear Chazy XII equation in terms of the classical theory of linear Fuchsian differential equations with three regular singular points, equivalently, via the hypergeometric equation. This is the main purpose of the present section.

Hypergeometric parametrization
In the following the domain D of the triangle functions s(z) will be taken as the interior of the orthogonal circle C discussed in Section 2.2, and the hypergeometric form of the Fuchsian differential equation (2.5) will be considered, in order to make contact with standard literature. If u(s) is a solution of (2.5), then the function satisfies the hypergeometric equation which can be written in more standard form as The transformation (4.1) sets the local exponents to (0, α) at s = 0, (0, β) at s = 1 and ( Note, however, that the exponent differences as well as the ratio z(s) of any two linearly independent solutions of (4.2b) coincide with those for (2.12). Consequently, one can employ the ratio of two independent solutions of (4.2a) instead of (2.5) to construct the conformal mapping and triangle function described in Section 2.2.
We next outline how to construct the triangle functions s(z) listed in Table 1 together with their orthogonal circles using pairs of linearly independent hypergeometric solutions (see, e.g., [24]). Notice from (2.12) that z(s) is defined up to an arbitrary Möbius transformation with three complex parameters of the z-plane depending on the choice of linearly independent solutions χ 1 and χ 2 of (4.2b). One can then choose two of the three parameters in the Möbius transformation in such a manner that the conformal map (2.12) results in a triangle which has the vertex z(0) placed at the origin of the z-plane, and the two circular arcs meeting there can be transformed to linear segments subtending angle πα at this vertex z(0). The remaining freedom in the Möbius transformation can be used to rotate the line segment connecting the vertices z(0) and z(1) onto the real axis. The remaining side joining z(1) and z(∞) is formed by the arc of a circle such that the origin O is in the exterior of this circle when the sum of the interior angles of the triangle is less than π, i.e., when α + β + γ < 1. Hence, it is possible to draw a line from the origin tangent to this circle at some point F (see Fig. 1, Appendix B). Then there exists a unique circle C with center at the origin O and passing through the point F, thus having radius OF, that is orthogonal to the straight edges of the triangle thus constructed. Consequently, C is orthogonal to each side of the triangle.
A pair of hypergeometric solutions whose ratio maps the upper half s-plane onto a triangle constructed above is given by χ 1 = 2 F 1 (a, b; c; s) and χ 2 = s 1−c 2 F 1 (a − c + 1, b − c + 1; 2 − c; s). Here the notation of (4.2b) has been used and 2 F 1 (a, b; c; s) is the standard hypergeometric series solution of (4.2b), analytic in the neighborhood of s = 0 with 2 F 1 (a, b; c; s = 0) = 1. Note that χ 2 vanishes at the branch point s = 0 if α = 1 − c > 0. The explicit form of the map is then .

(4.3)
Since the triples (α, β, γ) listed in Table 1  Re(s) > 0, then it follows from (4.3) that z(s) is also real and positive as s varies from 0 to 1 along the real s-axis. This confirms that one side of the triangle lies on the positive z-axis joining the vertices z(0) = 0 and z (1). The other side of the triangle originating from the vertex z(0) = 0 is the conformal image of the negative real s-axis on which the 2 F 1 functions are also real but the factor s 1−c = (−|s|) 1−c = |s|e iπα . This shows that the negative real s-axis is mapped to a linear segment joining z(0) and z(∞), and making an angle πα with the positive z-axis. One can also compute the vertices z(1) and z(∞) by considering the analytic continuations of the 2 F 1 functions into the neighborhoods of s = 1 and s = ∞. These are given by (see, e.g., [15,24]) where Γ(·) is the Gamma function. From the expressions for z(1) and z(∞), it is possible to determine the radius R of the orthogonal circle C which forms the natural barrier beyond which s(z) can not be analytically continued. In terms of the triple (α, β, γ) parameterizing the triangle function S(α, β, γ; z) the square of the radius of the barrier is [12] .
The above expression appears in [10,12] but without a derivation, which, although elementary, is not immediately obvious. For that reason, we have included a brief derivation in Appendix B.
The map z(s) in (4.3) is a Puiseux series in s of the form z(s) = s α ψ(s) where ψ(s) is analytic near s = 0 with ψ(0) = 0. In fact, the power series for ψ(s) can be readily derived from the series expansion of the 2 F 1 functions in (4.3). Since α = 1 n , n ≥ 2, n ∈ N, the series for z(s) can be inverted to obtain the power series of the inverse s(z) in the form where the coefficients b j can be obtained recursively from the coefficients in the series expansion of ψ(s). The series for s(z) converges in a neighborhood of z(0) = 0 and defines a single-valued, holomorphic function in this neighborhood as discussed earlier in Section 3.2. By analytic continuation of the hypergeometric functions onto the neighborhoods of s = 1 and s = ∞, it is possible to obtain similar series expansions for s(z) near z(1) and z(∞) as well. Note that s(z) has a pole of order 1 γ at z(∞). Let (χ 1 , χ 2 ) be the pair of hypergeometric solutions whose ratio defines z(s) as in (4.3), then s (z) = 1/z (s) = χ 2 1 /W (χ 1 , χ 2 ) where the Wronskian of the pair of solutions W (χ 1 , χ 2 ) = As α−1 (s − 1) β−1 from Abel's formula. The nonzero constant A is found by explicitly calculating the Wronskian of (χ 1 , χ 2 ) in (4.3) and letting s → 0. Thus one obtains A = (−1) 1−β α. Substituting the expression for s (z) in (3.2) yields a parametrization for y(z) in terms of χ 1 (s) and χ 1 (s), namely To be clear y(z) is expressed parametrically by z(s) in (4.3) and y(s) given by (4.4).
The Ramanujan-type triple (P ,Q,R) of automorphic functions introduced in Section 3.3 can also be formulated in terms of the hypergeometric function χ 1 (s) = 2 F 1 (a, b; c; s). The function P = y/(πi) is obtained directly from (4.4) above. In order to obtain expressions forQ,R, one first recalls from Section 3.1 that the quadratic differential Ω and its covariant derivatives are given in terms of s(z) and s (z) by where the rational functions V 2 (s) and V 3 (s) are obtained from (3.4). Then from (3.11) together with the relation s (z) = χ 2 1 /W (s) it follows that where W (s) = αs α−1 (1 − s) β−1 . The explicit form of the function V 2 (s) is given in (3.6), and from (3.4) one obtains V 3 (s) = V 2 (s) Alternatively, one could use (4.4) and the first two equations from (3.12) to derive (4.5).
Equations (4.4) and (4.5) provide parametrizations of the triple (P ,Q,R), equivalently the differential geometric quantities (η, Ω, ∇Ω) in terms of s and the hypergeometric function χ 1 (s). Conversely, it is also possible to derive expressions for S(α, β, γ; z) and χ 1 in terms of the functionsP ,Q andR. For instance, Case 1(a) in Table 1 gives rise to the following relations Note that if k → ∞ (Chazy III case with J = 12), one recovers the well known representation of the modular function S( 1 2 , 1 3 ; 0; z) from above in terms of the Ramanujan functions Q and R, which are the Eisenstein series of weight 4 and 6 respectively, for the modular group. Moreover, the first relation above leads to (a slight variant of) a remarkable identity discovered by Ramanujan [27] √ πQ This identity is obtained by first letting k → ∞ and then applying the Pfaff transformation (1 − s) 1/12 2 F 1 ( 1 12 , 1 12 ; 1 2 ; s) = 2 F 1 ( 1 12 , 5 12 ; 1 2 ; s s−1 ).

Pull-back maps of Schwarz functions
Recall that Table 2 of Section 3.3 lists the parametrizations of the Chazy XII solution y(z) with Chazy parameter 4 36−k 2 in terms of different Schwarz triangle functions. It follows from (3.2) that any two of these Schwarz functions are related via the differential relation φ 1 (z) = Cφ 2 (z) for some constant C, where the φ i (z) are listed in the last column of Table 2. In fact, the example given below Table 2 provides such a mapping between the triangle functions corresponding to Cases 1(a) and 1(b) of Table 2. In this subsection we systematically outline the transformations among the various triangle functions linking all the cases presented in Table 2.
The mappings of the Schwarz functions stem from the well-known algebraic transformations of the hypergeometric functions induced by the pull-back transformation of the corresponding hypergeometric differential equation (4.2b) [18]. These transformations are of the form Table 2. It suffices to derive the transformations relating Case 1(a) to all other cases in Table 2. Lets = θ(s) denote the rational map wheres = S( 1 2 , 1 3 , 1 k ; z) corresponds to Case 1(a) and s = S(α, β, γ; z) corresponds to any of the other cases. Lets = f 1 (s 1 ) = f 2 (s 2 ) denote two such rational maps for Schwarz functions s 1 (z) and s 2 (z) which parametrize the same Chazy XII solution y(z). Then the transformation between s 1 and s 2 can be expressed as s 2 = (f −1 2 • f 1 )(s 1 ), which may generally not be single valued.
Below we consider each case from Table 2 separately, treating Case (i) below as an illustrative example of the procedure.
Case (i). The mapping between the Schwarz functions S( 1 2 , 1 3 , 1 k ; z) and S( 1 3 , 1 3 , 2 k ; z) corresponding to Cases 1(a) and 1(b) of Table 2 follows from the well-known quadratic transformation [18] However, we do not pursue this direct approach here. We also remark that the differential φ(z)dz is a primitive of the incomplete beta integral (see, e.g., [15]) which is also related to the hypergeometric functions. Thus, the rational maps = θ(s) corresponds to algebraic transformation of the beta functions. The derivation of these rational maps directly from the solutions of (1.1) have not been studied to the best of our knowledge.

Concluding remarks
In this article, we have presented a method that converts the Chazy XII differential equation into an algebraic relation. This method is based on a simple differential geometric interpretation of the Chazy XII equation given by Dubrovin in [14]. By the way of elucidating this algebraic relationship, we have derived all possible parametrizations of the Chazy XII solution y(z) via the Schwarz triangle functions S(α, β, γ; z) for α + β + γ < 1. Our method can be extended in a straightforward way to the case when α + β + γ ≥ 1 but we do not pursue it here. Furthermore, we show that these parametrizations can be described in terms of classical hypergeometric functions. Using the hypergeometric theory, the center and radius of the natural barrier for the Chazy XII solution are found explicitly. Finally, the algebraic transformations of the hypergeometric functions are used to obtain rational maps between the Schwarz functions corresponding to the Chazy XII solution.
It is known that the Chazy III equation is equivalent to the Ramanujan differential relations (1.2) for the Eisenstein series P , Q, R for the modular group SL 2 (Z). Ramanujan also derived a number of remarkable identities among the functions P , Q, R using their representation in terms of hypergeometric functions. Likewise, we have introduced a Ramanujan-like triple (P ,Q,R) which satisfy differential relations that are equivalent to the Chazy XII equation. Additionally, these functions also satisfy interesting identities that can be derived from their hypergeometric representations. We have presented an example at the end of Section 4.1, but a comprehensive exploration of this line of investigation is for future study.
The rational maps between the Schwarz functions using the pull-backs of hypergeometric equations were presented in Section 4.2. However, we found that these maps can be also obtained by solving certain first order differential relations among the Schwarz functions. These relations follow directly from the representation of the Chazy XII solution y(z) in terms of the function φ(z) in (3.2). We believe that all rational maps of Schwarz functions satisfy such differential relations although we have not pursued this question further here since it is beyond the scope of this article. We plan to study this issue in a future work.

A Proof of Lemma 3.1
In this appendix we prove Lemma 3.1 from Section 3.2.

B Radius of the orthogonal circle
In this appendix we briefly outline a derivation of the expression in Section 4.1 for the radius R of the orthogonal circle C, which forms the natural barrier in the complex plane to the solutions of (1.1). Recall from Section 4.1 that (4.3) maps the upper-half s-plane onto a circular triangle with two straight edges denoted by OA and OB as shown in Fig. 1. The third side of the triangle is formed by the circular arc AB which is part of the circle Q with center at D, and radius AD = BD := r. The vertices O, A, B correspond to the image points z(0) = 0, z(1) and z(∞) of the map (4.3), respectively. The angle πα = ∠BOA, while angles πβ and πγ are the angles made by OA and OB respectively, with the circular arc AB. The orthogonal circle C is centered at the origin O and its radius OF:=R. Now consider the triangles OAB and DAB whose common side is given by the chord AB. From elementary considerations, it follows that ∠ADB = π − π(α + β + γ), ∠DAB = π( α+β+γ 2 ), and ∠OBA = π 2 − π( α+β−γ 2 ). Applying the law of sines to triangles OAB and DAB, and denoting OA = x, one finds that AB = x sin πα cos π( α+β−γ 2 ) , BD := r = AB sin π( α+β+γ 2 ) sin π(α + β + γ) .
Eliminating AB from above, yields the following expression for the radius of the circle Q r = x sin πα 2 cos π( α+β−γ 2 ) cos π( α+β+γ 2 ) . (B.1) Since the circles Q and C are mutually orthogonal, their radii OF and DF are perpendicular. Hence from the right triangle ODF with hypotenuse OD =: d it follows that R 2 = d 2 − r 2 . Next, considering the triangle OAD with ∠OAD = π 2 + πβ, one obtains d 2 = x 2 + r 2 + 2rx sin πβ. Substituting this into the expression for R 2 and using (B.1), one obtains after using some trigonometric identities. From (4.3) the line segment OA= x along the real axis is given by [15]. Substituting the above expression for x into (B.2), using the identity Γ(t)Γ(1 − t) = π csc(πt), and replacing (a, b, c) by their values in terms of (α, β, γ), one finally obtains the expression for R 2 in Section 4.1.