Integrable KdV Hierarchies on T^2=S^1\times S^1

Following our previous works on extended higher spin symmetries on the torus we focus in the present contribution to make a setup of the integrable KdV hierarchies on $T^{2} = S^{1} \times S^{1}$. Actually two particular systems are considered, namely the KdV and the Burgers non linear integrable model associated to currents of conformal weights (2, 2) and (1, 1) respectively. One key steps towards proving the integrability of these systems is to find their Lax pair operators. This is explicitly done and a mapping between the two systems is discussed.