  Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a
non-reflexive nodal rational plane curve of degree q+1 in characteristic p. We
investigate its automorphism group and defining equation. We also prove that
the surface obtained as the cyclic cover of the projective plane branched along
the Ballico-Hefez curve is unirational, and hence is supersingular. As an
application, we obtain a new projective model of the supersingular K3 surface
with Artin invariant 1 in characteristic 3 and 5.
