  For any smooth Hurwitz curve $\mathcal{H}_n: \, XY^n+YZ^n+X^nZ=0$ over the
finite field $\mathbb{F}_{p}$, an explict description of its Weierstrass points
for the morphism of lines is presented. As a consequence, the full automorphism
group ${\rm Aut}(\mathcal{H}_n)$, as well as the genera of all Galois subcovers
of $\mathcal{H}_n$, with $n\neq 3, p^r$, are computed. Finally, a question by
F. Torres on plane non nonsingular maximal curves is answered.
