Enumeration of Nonsingular Buekenhout Unitals

The only known enumeration of Buekenhout unitals occurs in the Desarguesian plane $PG(2,q^2)$.  In this paper we develop general techniques for enumerating the nonsingular Buekenhoutunitals embedded in any two-dimensional translation plane, and apply these techniques to obtain such an enumeration in the regular nearfield planes, the odd-order Hall planes, and the flag-transitive affine planes.  We also provide some computer data for small-order André planes of index two and give partial results toward an enumeration in this case