Finding integer solutions to systems of polynomial equations

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Not a full answer, but it seems that Sympy can find your special solution :

sage:  import sympy
sage: a, b = sympy.symbols("a, b", integer=True)
sage: A, B = sympy.symbols("A, B")
sage: f=A+B
sage: g=a*A+b*B
sage: sympy.solve(f-g, (a, b))
{a: 1, b: 1}

Now, your problem may have more solutions than that. Start with Maxima's solution (after a change of notations) :

sage: var("x, y, X, Y")
(x, y, X, Y)
sage: Sol=(X+Y-(x*X+y*Y)==0).solve((x, y)) ; Sol
[[x == -(Y*(r5 - 1) - X)/X, y == r5]]

A bit of algebraic mangling gives us :

sage: (Sol[0][0].subs(Sol[0][1].rhs()==Sol[0][1].lhs()).expand().collect_common_factors()-1).factor()
x - 1 == -Y*(y - 1)/X

which tells us that for any (integer) $y$, $(x,\, y)$ is a solution iif $\frac{Y(1-y)}{X}$ is integer. This is possible iif $\frac{X}{Y}$ is rational.

I believe that this describes the whole set of solutions, but I'd be glad to be demonstrated wrong.

Numerical example :

sage: (S:=(X+Y-(x*X+y*Y)==0).subs({X:5*sqrt(3), Y:3*sqrt(3)}).solve((x, y)))[0][0].subs(S[0][1].rhs()==S[0][1].lhs())
x == -3/5*y + 8/5

which shows that the non-trivial solutions are of the form $\left(\frac{-3k}{5},\,k\right),\,k\in\mathbb{Z}$;

HTH,

This isn't an answer, but one thing that could be useful in general with integers is to do

assume(A,'integer')

But solve in general is via Maxima, which will assume you are using "normal" real or complex variables, and this doesn't help (because Maxima explicitly avoids your assumptions in solving, because it is assumed you are using dummy variables in solve commands).

I am not really sure how the domain parameter influences things.