The Calogero–Bogoyavlenskii–Schiff Equation in $2+1$ Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the $(2+1)$-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this $(2+1)$-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the $(2+1)$-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.