  We consider mechanical systems on $T^*M$ with possibly irregular and
reducible first class contraints linear in the momenta, which thus correspond
to singular foliations on $M$. According to a recent result, the latter ones
have a Lie-infinity algebroid $(\cal M,Q)$ covering them, where we restrict to
the case of Lie-2 algebroids. We propose to consider $T^*\cal M$ as a potential
BFV extended phase space of the constrained system, such that the canonical
lift of the nilpotent vector field $Q$ yields automatically a solution to the
BFV master equation. We show that in this case, the BFV extension of the
Hamiltonian, providing a second corner stone of the BFV formalism, may be
obstructed. We identify the corresponding complex governing this second
extension problem explicitly (the first extension problem was circumvented by
means of the lift of the Lie-2 algebroid structure). We repeatedly come back to
the example of angular momenta on $T^*\mathbb R^3$: in this procedure, the
standard free Hamiltonian does not have a BFV extension -- while it does so on
$T^*(\mathbb R^3 \backslash \{0 \})$, with a relatively involved ghost
contribution singular at the origin.
