Let $y^2=xz-x^2+1$, and if value of $x$ is known, can $y$ and $z$ be directly calculated?

Question

Let $y^2=xz-x^2+1$, and if value of $x$ is known, can $y$ and $z$ be directly calculated? Given all variables are Integers.

Answer 1

$y^2=xz-x^2+1$

If $x$ is known, say $x=1$, then:

$$y^2=(1)z-(1)^2+1\implies y^2=z$$

$y$ and $z$ have infinitely many integer values they can be.

However, in another scenario where $x=0$, then:

$$y^2=0(z)-(0)^2+1 \implies y=\pm1$$

Here, $y$ has two values: $1$ and $-1$.

However, $z$ can have infinitely many integer values.

With these two above scenarios, the answer is sometimes that $y$ and $z$ have a finite amount of known values, other times, there can be infinitely many values plugged in, but the equation would still be true.