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We give examples of infinite order rational transformations that leave lineardifferential equations covariant. These examples are non-trivial yet simpleenough illustrations of exact representations of the renormalization group. Wefirst illustrate covariance properties on order-two linear differentialoperators associated with identities relating the same $_2F_1$ hypergeometricfunction with different rational pullbacks. We provide two new and more generalresults of the previous covariance by rational functions: a new Heun functionexample and a higher genus $_2F_1$ hypergeometric function example. We thenfocus on identities relating the same hypergeometric function with twodifferent algebraic pullback transformations: such remarkable identitiescorrespond to modular forms, the algebraic transformations being solution ofanother differentially algebraic Schwarzian equation that emerged in a paper byCasale. Further, we show that the first differentially algebraic equation canbe seen as a subcase of the last Schwarzian differential condition, therestriction corresponding to a factorization condition of some associatedorder-two linear differential operator. Finally, we also exploregeneralizations of these results, for instance, to $_3F_2$, hypergeometricfunctions, and show that one just reduces to the previous $_2F_1$ cases througha Clausen identity.  In a $_2F_1$ hypergeometric framework the Schwarzian condition encapsulatesall the modular forms and modular equations of the theory of elliptic curves,but these two conditions are actually richer than elliptic curves or $_2F_1$hypergeometric functions, as can be seen on the Heun and higher genus example.  This work is a strong incentive to develop more differentially algebraicsymmetry analysis in physics.

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics