Integral point sets over finite fields

We consider point sets in the affine plane ${\mathbb{F}}_q^2$ where each Euclidean distance of two points is an element of ${\mathbb{F}}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean spaces ${\mathbb{E}}^m$. We determine their maximal cardinality $\mathcal{I}({\mathbb{F}}_q,2)$. For arbitrary commutative rings $\mathcal{R}$ instead of ${\mathbb{F}}_q$ 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.