Chazy Classes IX-XI Of Third-Order Differential Equations

In this article, we study Classes IX–XI of the 13 classes introduced by Chazy (1911) in his classification of third‐order differential equations in the polynomial class having the Painlevé property. Classes IX and X are the only Chazy classes that have remained unsolved to this day, and they have been at the top of our “most wanted” list for some time. (There is an incorrect claim in the literature that these classes are unstable.) Here we construct their solutions in terms of hyperelliptic functions of genus 2, which are globally meromorphic. (We also add a parameter to Chazy Class X, overlooked in Chazy's original paper.) The method involves transforming to a more tractable class of fourth‐ and fifth‐order differential equations, which is the subject of an accompanying paper (paper I). Most of the latter equations involve hyperelliptic functions and/or higher‐order Painlevé transcendents. In the case of Chazy Class XI, the solution is elementary and well known, but there are interesting open problems associated with its coefficient functions, including the appearance of one of the aforementioned transcendents. In an appendix, we present the full list of Chazy equations (in the third‐order polynomial class) and the solutions of those that are not dealt with in the body of this article.