  We consider the subgroup of points of finite orbit through the action of an
endomorphism of a virtually free group, with particular emphasis on the
subgroup of eventually fixed points, EvFix($\varphi$): points whose orbit
contains a fixed point. We provide an algorithm to compute the subgroup of
fixed points of an endomorphism of a finitely generated virtually free group
and prove that finite orbits have cardinality bounded by a computable constant,
which allows us to solve several algorithmic problems: deciding if $\varphi$ is
a finite order element of End($G$), if $\varphi$ is aperiodic, if
EvFix($\varphi$) is finitely generated and, in the free group case, whether
EvFix($\varphi$) is a normal subgroup of $F_n$ or not. We also present a bound
for the rank of EvFix($\varphi$) in case it is finitely generated.
