  We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and
$f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients.
Moreover, we present results of computations of all integer points on certain
one parametric curves of genus 1 and 3, related to cubic and quartic fields,
respectively.
