Parameterizations of the Chazy equation

The Chazy equation $y''' = 2yy'' - 3y'^2$ is derived from the automorphic properties of Schwarz triangle functions $S(\alpha, \beta, \gamma; z)$. It is shown that solutions $y$ which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of $\alpha, \beta, \gamma$. The solutions are then systematically constructed. These analytic solutions provide all known and one new parametrization of the Eisenstein series $P, Q, R$ introduced by Ramanujan in his modular theories of signature 2, 3, 4 and 6.