  We introduce a new family of Poisson vertex algebras
$\mathcal{W}(\mathfrak{a})$ analogous to the classical $\mathcal{W}$-algebras.
The algebra $\mathcal{W}(\mathfrak{a})$ is associated with the centralizer
$\mathfrak{a}$ of an arbitrary nilpotent element in $\mathfrak{gl}_N$. We show
that $\mathcal{W}(\mathfrak{a})$ is an algebra of polynomials in infinitely
many variables and produce its free generators in an explicit form. This
implies that $\mathcal{W}(\mathfrak{a})$ is isomorphic to the center at the
critical level of the affine vertex algebra associated with $\mathfrak{a}$.
