  We investigate the total CR twist functional on transversal curves in the
standard CR 3-sphere $\mathrm S^3 \subset \mathbb C^2$. The question of the
integration by quadratures of the critical curves and the problem of existence
and properties of closed critical curves are addressed. A procedure for the
explicit integration of general critical curves is provided and a
characterization of closed curves within a specific class of general critical
curves is given. Experimental evidence of the existence of infinite countably
many closed critical curves is provided.
