  We extend our consideration of commutative subalgebras (rays) in different
representations of the $W_{1+\infty}$ algebra to the elliptic Hall algebra (or,
equivalently, to the Ding-Iohara-Miki (DIM) algebra
$U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$). Its advantage is that it
possesses the Miki automorphism, which makes all commutative rays equivalent.
Integrable systems associated with these rays become finite-difference and,
apart from the trigonometric Ruijsenaars system not too much familiar. We
concentrate on the simplest many-body and Fock representations, and derive
explicit formulas for all generators of the elliptic Hall algebra $e_{n,m}$. In
the one-body representation, they differ just by normalization from
$z^nq^{m\hat D}$ of the $W_{1+\infty}$ Lie algebra, and, in the $N$-body case,
they are non-trivially generalized to monomials of the Cherednik operators with
action restricted to symmetric polynomials. In the Fock representation, the
resulting operators are expressed through auxiliary polynomials of $n$
variables, which define weights in the residues formulas. We also discuss
$q,t$-deformation of matrix models associated with constructed commutative
subalgebras.
