On dominant rational maps from a very general complete intersection surface in P^4

Let S be a very general complete intersection surface of multidegree (d_1, d_2) in P^4. The following problem arises: determine the couples (d_1, d_2) such that the surface S does not have any "non-evident" rational map to other surfaces. By non-evident rational map, we mean nonbirational dominant map whose target space is not rational. We give a partial solution, presenting a class of multidegrees (d_1, d_2) which satisfy the above condition.