Imaginary-quadratic solutions of anti-Vandermonde systems in 4 unknowns and the Galois orbits of trees of diameter 4

The paper is devoted to an elementary Diophantine problem motivated by Grothendieck's dessins d'enfants theory. Namely, we consider the system of equations $ax^j+by^j+cz^j+dt^j=0$ ($j=1,2,3$) with natural $a$, $b$, $c$, and $d$. For trivial reasons it has no real (hence rational) nonzero solutions; we study the cases where it has imaginary quadratic ones. We suggest an infinite family of such cases covering all the imaginary quadratic fields. We discuss this result from the viewpoint of the Galois orbits of trees of diameter 4.