The authors study the parabolic operator \[ P_{\alpha} u = u_t - (|x|^{\alpha} u_x)_x \] for \(- 1 < x < 1\), and \(\alpha \in (0, 2)\). Note that this operator degenerates at the interior point \(x = 0\). The authors study the null controllability of \(P_{\alpha}\) for locally distributed controls acting only on one side of the origin (that is, on some interval \((a, b)\) with \(0 < a < b < 1\)). They prove that \(P\) is null controllable if and only if it is weakly degenerate, that is, \(\alpha \in (0, 1)\). So, in the strongly degenerate case \(\alpha \in [1, 2)\) in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy. This very interesting result is proved through spectral analysis and moment method. The authors also provide numerical evidence to illustrate their theoretical results.