  Let $X$ be a projective surface or a hyperk\"ahler manifold and $G \le
Aut(X)$. We give a necessary and sufficient condition for the existence of a
non-trivial $G$-equivariant fibration on $X$. We also show that two
automorphisms $g_i$ of positive entropy and polarized by the same nef divisor
are the same up to powers, provided that either $X$ is not an abelian surface
or the $g_i$ share at least one common periodic point. The surface case is
known among experts, but we treat this case together with the hyperk\"ahler
case using the same language of hyperbolic lattice and following Ratcliffe or
Oguiso.
  This arXiv version contains proofs omitted in the print version.
