  In this study, we give a survey of derivations of KdV-type equations with an
uneven bottom for several cases when small (perturbation) parameters $\alpha,
\beta, \delta$ are of different orders. Six different cases of such ordering
are discussed. Surprisingly, for all these cases the Boussinesq equations can
be made compatible only for the particular piecewise linear bottom profiles,
and the correction function has a universal form. For such bottom relief,
several new KdV-type wave equations are derived. These equations generalize the
KdV, the extended KdV (KdV2), the fifth-order KdV (KdV5) and the Gardner
equations.
