  We investigate a periodic version of the Benjamin-Ono (BO) equation
associated with a discrete Laplacian. We find some special solutions to this
equation, and calculate the values of the first two integrals of motion $I_1$
and $I_2$ corresponding to these solutions. It is found that there exists a
strong resemblance between them and the spectra for the Macdonald
$q$-difference operators. To better understand the connection between these
classical and quantum integrable systems, we consider the special degenerate
case corresponding to $q=0$ in more detail. Namely, we give general solutions
to this degenerate periodic BO, obtain explicit formulas representing all the
integrals of motions $I_n$ ($n=1,2,...$), and successfully identify it with the
eigenvalues of Macdonald operators in the limit $q\to 0$, i.e. the limit where
Macdonald polynomials tend to the Hall-Littlewood polynomials.
