  We consider finitely generated shift-invariant spaces (SIS) with additional
invariance in $L^2(\R^d)$. We prove that if the generators and their translates
form a frame, then they must satisfy some stringent restrictions on their
behavior at infinity. Part of this work (non-trivially) generalizes recent
results obtained in the special case of a principal shift-invariant spaces in
$L^2(\R)$ whose generator and its translates form a Riesz basis.
