Nearfield planes and the direction problem in $AG(2,q^2)$

We prove that the projective triangle of $PG(2,q^2), \ q$ odd, defines via indicator sets a regular nearfield spread of $PG(3,q),$ and conversely one of the indicator sets of such a spread is the projective triangle. Then we rephrase our results in the mainframe of the direction problem. Recall that if $U$ is a set of $s$ points in $AG(2,s),$ and $N$ is the number of the determined directions, when $s=p^2$ with $p$ an odd prime, G\'{a}cs, Lov\'{a}sz and Sz\H{o}nyi have proved that for $N=\frac{p^2+3}{2}$ there is a unique example and $U$ is affinely equivalent to the graph of the function $x\mapsto x^{\frac{p^2+1}{2}}.$ Here we prove a similar result for $s=q^2,$ \ $q$ any odd prime power, assuming some extra hypotheses.