  We study the relations governing the ring of quasiautomorphic forms
associated to triangle groups with a single cusp, thereby extending our earlier
results on Hecke groups. The Eisenstein series associated to these triangle
groups are shown to satisfy Ramanujan-like identities. These identities in turn
allow us to associate a nonlinear differential equation to each triangle group.
We show that they are solved by the quasiautomorphic weight-2 Eisenstein series
associated to the triangle group and its orbit under the group action. We
conclude by discussing the Painlev\'e property of these nonlinear differential
equations.
