Rational points on certain families of symmetric equations

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one. As an application, we explicitly describe $X_d(\mathbb{Q})$ for certain $d\geq3$, where $X_d: T_d(x)+T_d(y)=1$ and $T_d$ is the monic Chebychev polynomial of degree $d$. Moreover, we show how this later problem relates to orbit intersection problems in dynamics. Finally, we construct a new family of genus $3$ curves which break the Hasse principle, assuming the parity conjecture, by specifying our results to quadratic twists of $x^4-4x^2-4y^2+y^4=-6$.