Lax Pairs, Painleve ́ Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2 + 1)-Dimensional Equation

We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2 + 1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional CKdV equation. We show that the CKdV equation as well as the (2+1)-dimensional CKdV equation are integrable in the sense that they possess the Painleve ́ property. Some exact solutions are also constructed.