In this paper, we present a modified version with variable coefficients of the $(2+1)$ dimensional Korteweg-de Vries, or \textit{Calogero-Bogoyavlenskii-Schiff} equation, derived by applying the Painlevé test. Its Lax pair with a non-isospectral condition in $(2+1)$ dimensions is also given. Moreover a transformation which links the form with variable coefficients to the canonical one is shown.
