  Inspired by the recently published paper \cite{Hassainia-Hmidi}, the current
paper investigates the local well-posedness for the generalized $2d-$Boussinesq
system in the setting of regular/singular vortex patch. Under the condition
that the initial vorticity $\omega_{0}={\bf 1}_{D_0}$, with $\partial D_0$ is a
Jordan curve with a H\"older regularity $C^{1+\EE},\;0<\EE<1$ and the density
is a smooth function, we show that the velocity vector field is locally
well-posed and we also establish local-in-time regularity persistence of the
advected patch. Although, in the case of the singular patch, the analysis is
rather complicated due to the coupling phenomena of the system and the
structure of the patch. For this purpose, we must assume that the initial
nonlinear term is constant around the singular part of the patch boundary.
