Let $\text{q}\,\text{=}\,\text{2,}\,\text{3}$ and $f\left( X,\,Y \right)$, $g\left( X,\,Y \right)$, $h\left( X \right)$ be polynomials with integer coefficients. In this paper we deal with the curve
$f{{\left( X,\,Y \right)}^{\text{q}}}\,=\,h\left( X \right)g\left( X,\,Y \right)$
, and we show that under some favourable conditions it is possible to determine all of its rational points.