Note on an extension of the CDF equation to (2 + 1) dimensions

The Calogero—Degasperis—Fokas (CDF) equation is a (1 + 1)-dimensional integrable equation. We extend the CDF equation to (2+1) dimensions by a dimensional extension of the Lax pair. As a result, the higher-dimensional CDF equation and another higher dimensional equation are constructed. The Painlevé properties and (2 + 1)-dimensional traveling solitary solutions for the higher-dimensional equations obtained are also briefly reported. Moreover, the (2 + 1)-dimensional CDF equation can be reduced to some integrable equations.