Consider the diophantine equation $Q(x,y,z)=0$, where $x$, $y$ and $z$ are nonnegative integer unknowns and
$ Q(x,y,z)=x^3 + (-2y + 2)x^2 + ((z - 6)y + (2z + 1))x + ((2z - 4)y + 3z) $
Since the degree of $Q$ in $y$ and $z$ is $1$, the equation seems tractable. If $t\geq 0$, then $(x,y,z)=(t,t,t)$ is a solution. Are there any others ?