Equations of arbitrary order invariant under the Kadomtsev-Petviashvili symmetry group

By means of a simple new approach, a general Kadomtsev-Petviashvili (KP) family with an arbitrary function of group invariants of arbitrary order is proposed. It is proved that the general KP family possesses a common infinite dimensional Kac-Moody-Virasoro Lie point symmetry algebra. The known fourth order one can be re-obtained as a special example. The finite transformation group is presented in a clearer form. The Kac-Moody-Virasoro group invariant solutions and the Kac-Moody group invariant solutions of the KP family are determined by the Boussinesq and KdV families, respectively.