Let $a,b,c,d,s$ be distinct nonzero Eisenstein integers.
Claim : $$a^{11} + b^{11} = c^{11} + d^{11} = s$$
has only a finite number of solutions for any given $s$.
In fact at most $22$ for any $s$.
Is this true ? Is it known or studied ?
Any references ?
The equation,
$$a^{11}+b^{11} = c^{11}+d^{11}\tag1$$
is difficult to solve using quadratic integers, especially if we limit ourselves to Eisenstein integers. So we generalize things slightly and consider the ring of quadratic integers of $\mathbb{Q}(\sqrt{D})$ that is both a "principal ideal domain" and "Euclidean domain". Then for negative discriminants $D$ we only have five:
$$D = (-1,\,-2,\,-3,\,-7,\,-11)$$
and, voila, we have the single known identity for $11$th powers,
$$\left(\frac{1+\sqrt{-7}}2\right)^{11}+\left(\frac{1-\sqrt{-7}}2\right)^{11}=\left(\frac{1+\sqrt{-11}}2\right)^{11}+\left(\frac{1-\sqrt{-11}}2\right)^{11}= 67$$
given in this 2017 post. I don't know of any other solution for $(1)$ using quadratic integers, nor if,
$$a^{13}+b^{13} = c^{13}+d^{13}\tag2$$
is also solvable, but is probably not.