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We study various relations governing quasi-automorphic forms associated todiscrete subgroups of ${\rm SL}(2,\mathbb{R}) $ called Hecke groups. We showthat the Eisenstein series associated to a Hecke group ${\rm H}(m)$ satisfy aset of $m$ coupled linear differential equations, which are natural analoguesof the well-known Ramanujan identities for quasi-modular forms of ${\rmSL}(2,\mathbb{Z})$. Each Hecke group is then associated to a (hyper-)ellipticcurve, whose coefficients are determined by an anomaly equation. For the $m=3$and $4$ cases, the Ramanujan identities admit a natural geometricinterpretation as a Gauss-Manin connection on the parameter space of theelliptic curve. The Ramanujan identities also allow us to associate a nonlineardifferential equation of order $ m $ to each Hecke group. These equations arehigher-order analogues of the Chazy equation, and we show that they are solvedby the quasi-automorphic Eisenstein series $E_2^{(m)}$ associated to ${\rmH}(m) $ and its orbit under the Hecke group. We conclude by demonstrating thatthese nonlinear equations possess the Painlev\'e property.

Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations