A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably seems analytically intractable: Resolving $x$, $y$ into separate equations gives two irreducible polynomials of degree 12, with no evidently soluble roots (the same holds for $x+y$).
That situation suggested to me the following problem: What pairs of superellipses with integer coefficients have 'nice' intersections? The best case would be for there to be rational roots, but I'd be quite satisfied to see examples where the polynomials are reducible or the system is solvable.
EDIT) Let me narrow the scope of this: Consider \begin{cases} 3x^3+4y^3=a\\ 4x^4+3y^4=b \end{cases} for positive integers $a,b$. Certainly one can construct cases in which there are integer/rational/radical solutions. But suppose someone gives me integer $a$ and $b$. What criteria could I apply to see whether any 'nice' solutions exist?