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