  The KdV equation can be derived in the shallow water limit of the Euler
equations. Over the last few decades, this equation has been extended to
include both higher order effects (KdV2) and an uneven river bottom. Although
this equation is not integrable and has only one conservation law, exact
periodic and solitonic solutions exist for the even bottom case. The method
used to find them assumes the same function forms as for KdV solutions. KdV2
equation imposes more constraints on parameters of solutions. For soliton case
KdV2 solution occurs for particular ratio of wave amplitude to water depth
only. For periodic case physically relevant solutions are admissible only for
two narrow intervals of elliptic parameter $m$. For a range of $m$ near one the
cnoidal waves are upright as expected, but are inverted in $m$ region close to
zero. Properties of exact solutions of KdV and KdV2 are compared.
