The question must have meant you to solve T(S(x)) = x which should be simple enough.
It T is a translation then T(x) <> x for any x.
A rotation has one fixed point (or 2 if you count infinity as a point), so if you find the unique finite point where T(S(x)) you have found the centre of the rotation T(S)). Since the centre of a rotation is a fixed point.
The composition of a translation and a rotation is always another rotation (in either order)

The combination of rotation about two different points is also usually a rotation and again has one fixed point. In the exceptional cases where the equation has no solution you would find that it was a translation instead.