Affine ovoids and extensions of generalised quadrangles

A set a of vertices of a generalized quadrangle of order (s, t) is said to be a hyperoval if any line intersects a in either 0, or 2 points. A hyperoval Delta is called an affine ovoid if \Delta\ = 2st. It is well known that; mu-subgraphs in triangular extensions of generalized quadrangles are hyperovals. In the present paper we prove that if S is a triangular extension for GQ(s, t) with totally regular point graph Gamma such that mu = 2st, then s is even, Gamma is an r-antipodal graph of diameter 3 with r = 1 + s/2, and either s = 2, or t = s + 2.