If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the equation $$ yz=x^2 $$ are given by $x=uvw$, $y=uv^2$, $z=uw^2$ for some integers $u,v,w$. More formally, a subset $S \subset {\mathbb Z}^n$ is a polynomial family if there exists polynomials $P_1,\dots,P_n$ in some variables $u_1,\dots,u_k$ and integer coefficients such that $(x_1,\dots,x_n) \in S$ if and only if there exists integers $u_1,\dots,u_k$ such that $x_i=P_i(u_1,\dots,u_k)$ for $i=1,\dots,n$.

For some equations the solution set is not a polynomial family but is a finite union of polynomial families. The simplest example is the equation $xy=0$ with solutions $(x,y)=(u,0)$ and $(0,u)$.

In 2010, Vaserstein https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n2-p07-s.pdf answered a long-standing open question and showed that the solution set to the equation $$ xy - zt = 1 $$ is a polynomial family. As a corollary, he showed that the solutions to many equations, including $yz=x^2+a$ for any $a$ and $xy-zt=a$ for any $a$, are the finite unions of polynomial families.

The simplest examples not explicitly covered by this paper are equation $$ yz=x^2+x $$ and its variants $$ yz=x^2+x\pm 1. $$

The question is, for each of these equations, whether its solution set is a finite union of polynomial families? Or, in simple words, can we write down all solutions using polynomial expressions with parameters?

If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the equation $$ yz=x^2 $$ are given by $x=uvw$, $y=uv^2$, $z=uw^2$ for some integers $u,v,w$. More formally, a subset $S \subset {\mathbb Z}^n$ is a polynomial family if there exists polynomials $P_1,\dots,P_n$ in some variables $u_1,\dots,u_k$ and integer coefficients such that $(x_1,\dots,x_n) \in S$ if and only if there exists integers $u_1,\dots,u_k$ such that $x_i=P_i(u_1,\dots,u_k)$ for $i=1,\dots,n$.

For some equations the solution set is not a polynomial family but is a finite union of polynomial families. The simplest example is the equation $xy=0$ with solutions $(x,y)=(u,0)$ and $(0,u)$.

In 2010, Vaserstein https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n2-p07-s.pdf answered a long-standing open question and showed that the solution set to the equation $$ xy - zt = 1 $$ is a polynomial family. As a corollary, he showed that the solutions to many equations, including $yz=x^2+a$ for any $a$ and $xy-zt=a$ for any $a$, are the finite unions of polynomial families. As noted by Fedor Pertov in the comment, this result also implies parametrization of the solution set of the equation $$ yz=x^2+x. $$

The simplest examples that seems to be not explicitly covered by this paper are equations $$ yz=x^2+x+1 $$ and $$ yz=x^2+x-1. $$

The question is, for each of these equations, whether its solution set is a finite union of polynomial families? Or, in simple words, can we write down all solutions using polynomial expressions with parameters?