Birational transformations preserving rational solutions of algebraic ordinary differential equations

We characterize the set of all rational transformations with the property of preserving the existence of rational solutions of algebraic ordinary differential equations (AODEs). This set is a group under composition and, by its action, partitions the set of AODEs into equivalence classes for which the existence of rational solutions is an invariant property. Moreover, we describe how the rational solutions, if any, of two different AODEs in the same class are related.