How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.?
As a specific example, do there exist integers $x$ and $y$ such that $$ x^3+x^2y-y^3-y+3 = 0 ? $$
As noted in the previous Mathoverflow question Integer points of an elliptic curve (and in the comments to this question) there is a paper
Stroeker and de Weger's "Solving elliptic diophantine equations: The general cubic case."
(see also Stroeker, Roel, and Nikolaus Tzanakis. "Computing all integer solutions of a genus 1 equation." Mathematics of computation 72.244 (2003): 1917-1933. for higher degree equations)
that describes the algorithm, but the details are quite non-trivial to work out by hand, especially because I need to solve many equations in this form.
Interestingly, in the more difficult case of genus 2 equations there is a Magma code, see e.g.
https://math.stackexchange.com/questions/2700512/is-x5x5-y2-solvable-over-the-integers/2700727
and
Find all rational solutions of $x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1)$
that, for many specific equations (although not in general) able to automatically find all rational (and therefore all integer) solutions. Hence, it is a bit surprise for me that integer points on genus 1 curve (including Elliptic curve given by cubic equation in non-Weierstrass form) turned out to be more difficult for implementation.
(Remark: The question has been revised in the light of comments of Chris Wuthrich that you can find below).