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Following our previous works on extended higher spin symmetries on the toruswe focus in the present contribution to make a setup of the integrable KdVhierarchies on $T^{2} = S^{1} \times S^{1}$. Actually two particular systemsare considered, namely the KdV and the Burgers non linear integrable modelassociated to currents of conformal weights (2, 2) and (1, 1) respectively. Onekey steps towards proving the integrability of these systems is to find theirLax pair operators. This is explicitly done and a mapping between the twosystems is discussed.

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