We consider KdV currents in a quantum field theory obtained by deforming a two dimensional conformal field theory via the irrelevant operator . In this paper we determine their torus one–point functions modular properties. We find that the one–point functions decompose into a direct sum of two non–holomorphic modular forms. We also obtain a general differential equation that the KdV generalized torus partition sum satisfies. The differential equation provides a non–perturbative description of deformed theories.