  For the generalized $p$-power Korteweg-de Vries equation, all non-periodic
travelling wave solutions with non-zero boundary conditions are explicitly
classified for all integer powers $p\geq 1$. These solutions are shown to
consist of: bright solitary waves and static humps on a non-zero background for
odd $p$; dark solitary waves on a non-zero background and kink waves for even
$p$ in the defocusing case; pairs of bright/dark solitary waves on a non-zero
background, and also bright and dark heavy-tail waves (with power decay) on a
non-zero background, for even $p$ in the focusing case. An explicit physical
parameterization is given for each type of solutionin terms of the wave speed
$c$, background size $b$, and wave height/depth $h$. The allowed kinematic
region in $(c,b)$ as well as in $(h,b)$ for existence of the solutions is
derived, and other main kinematic features are discussed. Explicit formulas are
presented in the integrable cases $p=1,2$, and in the higher power cases
$p=3,4$.
