Consider the diophantine equation three variables $x,y,z$

$$x^2+L(y,z)x+Q(y,z)=0$$ where $L(y,z)$ is a linear polynomial in $y,z$ and $Q(y,z)$ is a quadratic polynomial in $y,z$.

In general such an equation is difficult to solve if $x,y,z$ are independent.

However suppose that there are univariate linear functions $L'(y)$ and $L''(z)$ such that if $(x^*,y^*,z^*)$ is a solution to the three variable diophantine equation then

1. $GCD(L'(y^*),L''(z^*)=1$

2. There exists integers $1<a,b$ with $a|L'(y^*)$ and $b|L''(z^*)$ such that $ab=x^*$

holds always. Then the unknown $x$ depends on $y,z$ and so in principle we should be able to eliminate $x$ and make this as a two variable quadratic diophantine equation.

1. One cannot say there is no small degree algebraic relation between $x$ and $y,z$ since we have already provided one by the quadratic equation. Unless we provide a different relation we cannot use elimination theory. However there is arithmetic relation. There is some hope. Is there a way to at least in principle reduce the problem to solving two unknowns through some functions coming from arithmetic?

2. What is the minimal degree of any other polynomial relation between $x,y,z$?