  For any simple Lie algebra $\mathfrak{g}$ and an element
$\mu\in\mathfrak{g}^*$, the corresponding commutative subalgebra
$\mathcal{A}_{\mu}$ of $\mathcal{U}(\mathfrak{g})$ is defined as a homomorphic
image of the Feigin-Frenkel centre associated with $\mathfrak{g}$. It is known
that when $\mu$ is regular this subalgebra solves Vinberg's quantisation
problem, as the graded image of $\mathcal{A}_{\mu}$ coincides with the
Mishchenko-Fomenko subalgebra $\overline{\mathcal{A}}_{\mu}$ of
$\mathcal{S}(\mathfrak{g})$. By a conjecture of Feigin, Frenkel and Toledano
Laredo, this property extends to an arbitrary element $\mu$. We give sufficient
conditions which imply the property for certain choices of $\mu$. In
particular, this proves the conjecture in type C and gives a new proof in type
A. We show that the algebra $\mathcal{A}_{\mu}$ is free in both cases and
produce its generators in an explicit form. Moreover, we prove that in all
classical types generators of $\mathcal{A}_{\mu}$ can be obtained via the
canonical symmetrisation map from certain generators of
$\overline{\mathcal{A}}_{\mu}$. The symmetrisation map is also used to produce
free generators of nilpotent limits of the algebras $\mathcal{A}_{\mu}$ and
give a positive solution of Vinberg's problem for these limit subalgebras.
