  We study integrable hierarchies associated with spectral problems of the form
$P\psi=\lambda Q\psi$ where $P,Q$ are difference operators. The corresponding
nonlinear differential-difference equations can be viewed as inhomogeneous
generalizations of the Bogoyavlensky type lattices. While the latter turn into
the Korteweg--de Vries equation under the continuous limit, the lattices under
consideration provide discrete analogs of the Sawada--Kotera and
Kaup--Kupershmidt equations. The $r$-matrix formulation and several simplest
explicit solutions are presented.
