  On the generalized tangent bundle of a smooth manifold, we study
skew-symmetric endomorphism satisfying an arbitrary polynomial equation with
real constant coefficients. We study the compatibility of these structures with
the de Rham operator and the Courant-Dorfman bracket. In particular we isolate
several conditions that when restricted to the motivating example of
generalized almost complex structure are equivalent to the notion of
integrability.
