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We show that non-linear Schwarzian differential equations emerging fromcovariance symmetry conditions imposed on linear differential operators withhypergeometric function solutions, can be generalized to arbitrary order lineardifferential operators with polynomial coefficients having selecteddifferential Galois groups. For order three and order four linear differentialoperators we show that this pullback invariance up to conjugation eventuallyreduces to symmetric powers of an underlying order-two operator. We give,precisely, the conditions to have modular correspondences solutions for suchSchwarzian differential equations, which was an open question in a previouspaper. We analyze in detail a pullbacked hypergeometric example generalizingmodular forms, that ushers a pullback invariance up to operator homomorphisms.We expect this new concept to be well-suited in physics and enumerativecombinatorics. We finally consider the more general problem of the equivalenceof two different order-four linear differential  Calabi-Yau operators up to pullbacks and conjugation, and clarify the caseswhere they have the same Yukawa couplings.

Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)