  The Painlev\'e property for a (2+1)-dimensional Korteweg-de Vries (KdV)
extension, the combined KP3 (Kadomtsev- Petviashvili) and KP4 (cKP3-4) is
proved by using Kruskal's simplification. The truncated Painlev\'e expansion is
used to find the Schwartz form, the B\"acklund/Levi transformations and the
residual nonlocal symmetry. The residual symmetry is localized to find its
finite B\"acklund transformation. The local point symmetries of the model
constitute a centerless Kac-Moody-Virasoro algebra. The local point symmetries
are used to find the related group invariant reductions including a new Lax
integrable model with a fourth order spectral problem. The finite
transformation theorem or the Lie point symmetry group is obtained by using a
direct method.
