  We introduce a free field realization of the central extension of the Lie
algebra $\mathcal{D}_q$ of difference operators on the circle in terms of the
fermionic $\eta$-$\xi$ system. This realization admits a nontrivial Jordan
block structure. We also review the free field realization of
$\mathcal{W}_{1+\infty}$ algebra, and point out some relations beween its
generators of weight zero and the local integrals of motion of Bazhanov
Lukyanov and Zamolodchikov. Finally we compute the finitized characters, and
the continuum characters of the Local Integrals of Motion, and find out and
interesting analogy with the generating functions for the counting of branched
covers of elliptic curves.
