Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of Rankin-Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin-Cohen structure. In the second part of the paper we consider triangle groups $\mathrm{\Delta }(n,m,\mathrm{\infty })$. By means of modular embeddings, we associate to every such group a number of systems of non linear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on $\mathrm{\Delta }(n,m,\mathrm{\infty })$ are generated by the solutions of one such system (which is of Rankin-Cohen type). As a corollary we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non classical setting, we construct the space of integral weight twisted modular form on $\mathrm{\Delta }(2,5,\mathrm{\infty })$ from solutions of systems of nonlinear ODEs.