  We solve the equivariant generalized Nash problem for any non-rational normal
variety with torus action of complexity one. Namely, we give an explicit
combinatorial description of the Nash order on the set of equivariant
divisorial valuations on any such variety. Using this description, we
positively solve the classical Nash problem in this setting, showing that every
essential valuation is a Nash valuation. We also describe terminal valuations
and use our results to answer negatively a question of de Fernex and Docampo by
constructing examples of Nash valuations which are neither minimal nor
terminal, thus illustrating a striking new feature of the class of
singularities under consideration.
