As the title says. Which finite projective planes admit a symmetric incidence matrix? I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F_q)$ can always have a symmetric matrix, but he was unsure about the non-Desarguesian planes.

I am mostly asking for reference (both for the Desarguesian and non-Desaruesian case), but any insight or (quick) proof is welcome.