  We define a Gauss map for surfaces in the universal cover of the Lie group
PSL_2(R) endowed with a left-invariant Riemannian metric having a 4-dimensional
isometry group. This Gauss map is not related to the Lie group structure. We
prove that the Gauss map of a nowhere vertical surface of critical constant
mean curvature is harmonic into the hyperbolic plane H^2 and we obtain a
Weierstrass-type representation formula. This extends results in H^2 x R and
the Heisenberg group Nil_3, and completes the proof of existence of harmonic
Gauss maps for surfaces of critical constant mean curvature in any homogeneous
manifold diffeomorphic to R^3 with isometry group of dimension at least 4.
