Satellite (1,1)-knots and meridionally incompressible surfaces

Let K be a satellite (1,1)-knot in S3 and T an essential torus in the exterior of K. Suppose that there is a closed orientable essential meridionally incompressible surface, say F. We show that F is always isotopic to the essential torus T. To this end, we study compact orientable essential surfaces in the exterior of non-trivial 2-bridge links. Moreover, we characterize (1,1)-splittings of (1,1)-knots containing F in their exterior by using the concept of the distance of (1,1)-splittings.