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In this work we shall study the well-posedness and ill-posedness of theCauchy problem associated to the equation \begin{equation*}u_{t}+a(u^{n})_{x}+(b\mathscr{H} u_{t}+u_{yy})_{x}=0, \end{equation*} inanisotropic weigthed Sobolev spaces. In particular, we examine the uniquecontinuation property for solutions, of this, with polynomial decay.

On the Cauchy problem associated to a regularized Benjamin-Ono--Zakharov-Kuznetsov (rBO-ZK) type equation