  We extend the notion of T-duality to manifolds endowed with non-principal
torus actions. The singularities of the torus action are controlled by a
certain Lie algebroid, called the elliptic tangent bundle. Using this Lie
algebroid, we explain how certain invariant generalized complex structures can
be transported via T-duality. Along the way, we use the elliptic tangent bundle
to define connections for these torus action, and give new insight to the
classification of such actions by Haefliger-Salem.
