  We find classes of projective manifolds that are elliptic in the sense of
Gromov and such that the affine cones over these manifolds also are elliptic
off their vertices. For example, the latter holds for any generalized flag
manifold of dimension at least 3 successively blown up in a finite set of
points and infinitesimally near points. This also holds for any smooth
projective rational surface. For the affine cones, the Gromov ellipticity is a
much weaker property than the flexibility. Nonetheless, it still implies the
infinite transitivity of the action of the endomorphism monoid on these cones.
