  We investigate the transitivity properties of the group of morphisms
generated by Vieta involutions on the solutions in congruences to the Markoff
equation as well as to other Markoff type affine cubic surfaces. These are
dictated by the finite $\bar{\mathbb Q}$ orbits of these actions and these can
be determined effectively. The results are applied to give forms of strong
approximation for integer points, and to sieving, on these surfaces
