CONO-SUR 1991 Problem 3

It is known that the number of real solutions of the following system if finite. Prove that this system has an even number of solutions:

$(y^2+6)(x-1)=y(x^2+1)$

$(x^2+6)(y-1)=x(y^2+1)$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.