Parameterizations of the Chazy equation

The Chazy equation $y^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=2y{y}^{\mathrm{\prime }\mathrm{\prime }}-3y^{\mathrm{\prime }2}$ 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.