On a Baer involution which maps an oval into itself (Q797144)

Let \({\mathcal O}\) be an oval in a finite projective plane \(\pi\) of odd order. We derive two results relating to a Baer involution which maps \({\mathcal O}\) into itself without fixing any point of \({\mathcal O}\). (1) We prove that such a Baer involution cannot exist if \({\mathcal O}\) has certain special properties. (2) We prove, by showing that every oval in a finite Moufang plane of odd order has these special properties, that \(\pi\) cannot be a Moufang plane if it does admit such a Baer involution.