$C$-KP and $B$-KP Equations Related to the Generalized Quartic Hénon–Heiles Hamiltonian

In their classification of soliton equations from a group theoretical standpoint according to the representation of infinite Lie algebras, Jimbo and Miwa listed bilinear equations of low degree for the KP and the modified KP hierarchies. In this list, we consider the $(1+1)$-dimensional reductions of three particular equations of special interest for establishing some new links with the generalized Hénon–Heiles Hamiltonian, possibly useful for integrating the latter with functions having the Painlevé property. Two of those partial differential equations have $N$-soliton solutions that, as for the Kaup–Kupershmidt equation, can be written as the logarithmic derivative of a Grammian. Moreover, they can describe head-on collisions of solitary waves of different type and shape.