 Solution

Suppose there were no singular point in the Riemann sphere. Then every finite point would be ordinary, so $p$ and $q$ would be entire functions. Ordinary behavior at infinity would force $p(z)=2/z+O(z^{-2})$ and $q(z)=O(z^{-4})$. In particular $p(z)\to 0$, so Liouville theorem would give $p\equiv 0$, contradicting the required leading term $2/z$. Thus infinity cannot also be ordinary, and every such differential equation has at least one singular point in $\overline{\mathbb{C}}$.