THE PROBLEM: to find all translation planes A of order 81 which admit two collineations σ and τ of order 3 such that the fixed-point sets F(σ) and F(τ) are Baer subplanes (i.e. σ and τ are Baer 3-collineations) which properly overlap,

0 ≠ F(σ) ∩ F(τ) ≠ F(σ).

If A is a translation plane of characteristic p > 3, then it is known [2] that no such overlapping Baer p-collineations exist. However, the nearfield plane of order 9 does admit such collineations. The general problem then is the investigation of overlapping Baer 3-collineations in translation planes of order 32e, for e > 1.

THEOREM: There are (up to isomorphism) n planes of order 81 admitting such overlapping 3-collineations σ and τ, where 3 ≤ n ≤ 6. (Note: all examples but not necessarily all isomorphisms, are known.)

PROOF: Normalize extensively by hand; then compute all normalized cases by machine. In stage 1 of the computing, 96 × 812 cases were checked, resulting in approximately 5,000 successes. During stage 2, for each success in stage 1, approximately 100 cases were checked. Each of the approximately two dozen successes in stage 2 describes a plane of the given type, but there are many obvious isomorphisms, resulting in 3 to 6 isomorphism classes.