  For any involution $\sigma$ of a semisimple Lie algebra $\mathfrak g$, one
constructs a non-reductive Lie algebra $\mathfrak k$, which is called a
$\mathbb Z_2$-contraction of $\mathfrak g$. In this paper, we attack the
problem of describing maximal commutative subalgebras of the Poisson algebra
$S(\mathfrak k)$. This is closely related to the study of the coadjoint
representation of $\mathfrak k$ and the set, $\mathfrak k^*_{reg}$, of the
regular elements of $\mathfrak k^*$. By our previous results, in the context of
$\mathbb Z_2$-contractions, the argument shift method provides maximal
commutative subalgebras of $S(\mathfrak k)$ whenever $codim(\mathfrak k^*
\setminus \mathfrak k^*_{reg})\ge 3$.
  Our main result here is that $codim(\mathfrak k^* \setminus \mathfrak
k^*_{reg})\ge 3$ if and only if the Satake diagram of $\sigma$ has no trivial
nodes. (A node is trivial, if it is white, has no arrows attached, and all
adjacent nodes are also white.) The list of suitable involutions is provided.
We also describe certain maximal commutative subalgebras of $S(\mathfrak k)$ if
the (-1)-eigenspace of $\sigma$ in $\mathfrak g$ contains regular elements.
