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.

There is no algebraic relation between $x$ and $y,z$ and so 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?