a2+b2+c2+1=k abca^2+b^2+c^2+1 = k\,abca2+b2+c2+1=kabc

has a solution in positive integers $a,b,c$. For that $k$, prove every solution has $a,b,c$ all odd, and that every solution can be reduced to $(1,1,1)$ by repeated moves $(a,b,c)\mapsto(a,b,4ab-c)$.