  We consider the problem of a self-interacting scalar field nonminimally
coupled to the three-dimensional BTZ metric such that its energy-momentum
tensor evaluated on the BTZ metric vanishes. We prove that this system is
equivalent to a self-dual system composed by a set of two first-order
equations. The self-dual point is achieved by fixing one of the coupling
constant of the potential in terms of the nonminimal coupling parameter. At the
self-dual point and up to some boundary terms, the matter action evaluated on
the BTZ metric is bounded below and above. These two bounds are saturated
simultaneously yielding to a vanishing action for configurations satisfying the
set of self-dual first-order equations.
