Expansions of O-minimal Structures on the Real Field by Trajectories of Linear Vector Fields

Let R be an o-minimal expansion of the field of real numbers that defines nontrivial arcs of both the sine and exponential functions. Let G be a collection of images of solutions on intervals to differential equations y′ = F (y), where F ranges over all R-linear transformations R → R. Then either the expansion of R by the elements of G is as well behaved relative to R as one could reasonably hope for, or it defines the set of all integers Z, and thus is as complicated as possible. In particular, if R defines any irrational power functions, then the expansion of R by the elements of G either is o-minimal or defines Z.