Existing results on the characterization of stable distributions lead one to expect that \[ E| tX+uY|^ p=A(| t|^ q+| u|^ q)^{p/q}\quad\text{for all } t,u\in\mathbb{R} \] with some \(A>0\), \(X\) and \(Y\) independent and identical in distribution, implies that \(X\) has a stable distribution. The author showed in a previous paper that this does not hold and now continues to show that even under additional restrictions on the behaviour of \(P(| X|\geq x)\), \(x\to\infty\), such a characterization is not valid. It is shown that, if the above equation holds for \(q=2\) and two different positive values of \(p\neq 2m\) for integer \(m\), then \(X\) has a normal distribution.