Integral automorphisms of affine planes over finite fields

A permutation of the affine plane Fq2 is called an integral automorphism if it preserves the integral distance defined on Fq2. In [7] M. Kiermaier and S. Kurz described (q(q−1)r)2 integral automorphisms of Fq2, where q=pr, p is a prime, and q≡1(mod 4), and also conjectured that these comprise all integral automorphisms if q∉{5,9}. In this paper we prove the conjecture, and by this complete the classification of integral automorphisms of affine planes over finite fields. Our proof relies on various results about primitive permutation groups, including the classification of finite primitive affine permutation groups of rank 3.