  We study mechanical models with first class constraints, not restricting to
only regular and irreducible ones. We show that while the BFV charge always
exists, the BFV extension of the Hamiltonian may be obstructed. Somewhat
astonishingly this happens already for a free particle in $\mathbb{R}^3$ when
constraining its angular momentum. We determine a complex whose cohomology
governs the extendability and provide examples with interesting geometrical
extensions.
