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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 theCKdV equation, in the search of a (2+1)-dimensional case and thereby propose anew equation in (2+1) dimensions. We named this the (2+1)-dimensional CKdVequation. We show that the CKdV equation as well as the (2+1)-dimensional CKdVequation are integrable in the sense that they possess the Painlev\'e property.Some exact solutions are also constructed

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