On three-dimensional constant curvature strings

I consider a string theory with target space three-dimensional Euclidean space whose action, besides the standard area term, contains another one proportional to the volume enclosed by the surface. The associated variational problem yields as solutions constant mean curvature surfaces. The equation of motion can be rewritten as a zero curvature condition for a SU(2) connection, thus implying the existence of an infinite set of conserved nonlocal charges. I also show how a description of the Gauss map of the surface in terms of SU(2) spinors allows for yet a different description of this result by means of a Gross-Neveu spinorial model coupled to 2-D gravity.