  Nonlinear generalizations of integrable equations in one dimension, such as
the KdV and Boussinesq equations with $p$-power nonlinearities, arise in many
physical applications and are interesting in analysis due to critical
behaviour. This paper studies analogous nonlinear $p$-power generalizations of
the integrable KP equation and the Boussinesq equation in two dimensions.
Several results are obtained. First, for all $p\neq 0$, a Hamiltonian
formulation of both generalized equations is given. Second, all Lie symmetries
are derived, including any that exist for special powers $p\neq0$. Third,
Noether's theorem is applied to obtain the conservation laws arising from the
Lie symmetries that are variational. Finally, explicit line soliton solutions
are derived for all powers $p>0$, and some of their properties are discussed.
