In this paper we consider the gathering of oblivious mobile robots in a n-node ring. In this context, the single class of configurations left open in the most recent study [2] is SP4 (a special class of configurations with only four robots). We present an algorithm to solve some of the most intricate configurations in SP4, those that can lead to a change of the axis of symmetry. Our approach lays the methodological bases for closing the remaining open cases for SP4 solvability.