Intransitive collineation groups of ovals fixing a triangle

We investigate collineation groups of a finite projective plane of odd order fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which there exists a fixed triangle off the oval is considered in detail. Our main result is the following. Theorem. Let P be a finite projective plane of odd order n containing an oval O. If a collineation group G of P satisfies the properties: (a) G fixes O and the action of G on O yields precisely two orbits O_1 and O_2, (b) G has even order and a faithful primitive action on O_2, (c) G fixes neither points nor lines but fixes a triangle ABC in which the points A, B, C are not on the oval O, then n is an element of {7, 9, 27}, the orbit O_2 has length 4 and G acts naturally on O_2 as A_4 or S_4. Each order n in {7, 9, 27} does furnish at least one example for the above situation; the determination of the planes and the groups which do occur is complete for n = 7, 9; the determination of the planes is still incomplete for n = 27.