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Solve equations integrally

Is it possible to solve polynomial equations over the integers in Sage?

For example, I want to find the integral solutions of x*y + y*z + z*x = 7 and x + y + z = 5. One possible solution is [x = 1, y = 1, z = 3], and there are only finitely many. Geometrically, I want to obtain the rational points on the intersection of a quadratic hypersurface and a couple of hyperplanes.

After some testing I have come to the conclusion that Maple cannot do this, but Mathematica does seem to be able to produce those solutions. However, I'm new to Sage and don't know if this is within its reach.

I would be very grateful for your help