  We give examples of infinite order rational transformations that leave linear
differential equations covariant. These examples are non-trivial yet simple
enough illustrations of exact representations of the renormalization group. We
first illustrate covariance properties on order-two linear differential
operators associated with identities relating the same $_2F_1$ hypergeometric
function with different rational pullbacks. We provide two new and more general
results of the previous covariance by rational functions: a new Heun function
example and a higher genus $_2F_1$ hypergeometric function example. We then
focus on identities relating the same hypergeometric function with two
different algebraic pullback transformations: such remarkable identities
correspond to modular forms, the algebraic transformations being solution of
another differentially algebraic Schwarzian equation that emerged in a paper by
Casale. Further, we show that the first differentially algebraic equation can
be seen as a subcase of the last Schwarzian differential condition, the
restriction corresponding to a factorization condition of some associated
order-two linear differential operator. Finally, we also explore
generalizations of these results, for instance, to $_3F_2$, hypergeometric
functions, and show that one just reduces to the previous $_2F_1$ cases through
a Clausen identity.
  In a $_2F_1$ hypergeometric framework the Schwarzian condition encapsulates
all 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 algebraic
symmetry analysis in physics.
