Let X be a 3-dimensional affine variety with a faithful action of a 2-dimensional torus T. Then the space of first order infinitesimal deformations T¹(X) is graded by the characters of T, and the zeroth graded component T¹(X)₀ consists of all equivariant first order (infinitesimal) deformations.

Suppose that using the standard construction of such varieties, one can obtain X from a proper polyhedral divisor D on P¹ such that the tail cone of (any of) the used polyhedra is pointed and full-dimensional, and all vertices of all polyhedra are lattice points. Then we compute dim T¹(X)₀ and find a formally versal equivariant deformation of X. We also establish a connection between our formula for dim T¹(X)₀ and known formulas for the dimensions of the graded components of T¹ of toric varieties.