  For a regular functional J defined on a Hilbert space X, we consider the set
N of points x of X such that the projection of the gradient of J at x onto a
closed linear subspace V(x) of X vanishes. We study sufficient conditions for a
constrained critical point of J restricted to N to be a free critical point of
J, providing a unified approach to different natural constraints known in the
literature, such as the Birkhoff-Hestenes natural isoperimetric conditions and
the Nehari manifold. As an application, we prove multiplicity of solutions to a
class of superlinear Schr\"odinger systems on singularly perturbed domains.
