  We prove an equivariant implicit function theorem for variational problems
that are invariant under a varying symmetry group (corresponding to a bundle of
Lie groups). Motivated by applications to families of geometric variational
problems lacking regularity, several non-smooth extensions of the result are
discussed. Among such applications is the submanifold problem of deforming the
ambient metric preserving a given variational property of a prescribed family
of submanifolds, e.g., constant mean curvature, up to the action of the
corresponding ambient isometry groups.
