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Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic integers, while Sage solves $x+y=1$ with $x,y$ $S$-integers, but only for $S$ the set of primes over a fixed rational primes. Is that true? Is there any other implementation available?

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This SageMath implementation promises the full generality you are seeking:

A robust implementation for solving the S-unit equation and several applications

See also this Phys.Org announcement.

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    $\begingroup$ Thanks, that is true indeed. I figured out the problem is that I was using the code K<i>=QuadraticField(-1) instead of K<i>=NumberField(x^2+1), and using the former it gives an error if you try to compute solutions for S a set of primes of different residue characteristic. $\endgroup$
    – Ferra
    Commented Nov 20, 2020 at 10:11

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