Integral point sets over finite fields

We consider point sets in the affine plane F-q(2) where each Euclidean distance of two points is an element of F-q. These sets are called integral point sets and were originally defined in m- dimensional Euclidean spaces Em. We determine their maximal cardinality I(F-q, 2). For arbitrary commutative rings R instead of Fq or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.