On certain symmetry reduction systems of the three-wave resonant interaction in (2+1) dimensions

We study three systems of nonlinear differential equations obtained from the symmetry reduction of the three-wave resonant interaction (2D-3WR) in (2+1) dimensions. We show that two of such systems are reducible to Painlevé III and Painlevé VI equations, respectively, while for the third system a spectral problem is found via a prolongation tecnique. New solutions of the 2D-3WR equations can be determined through the relationships existing between the reduction variables and the original three-wave resonant fields.