One of the forms in which the Diophantine equation in question can be found in the literature is this:

Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in positive integers $x, y$, and $z$.

There are some other variants of it here and there. It is usually given once the instructor has touched upon the Jacobi symbol.

My questions about the problem are the following two:

Do you happen to know where it was that the problem first appeared? If I understand correctly, the variation in which one is asked to establish that $z^{2} = 4xyt^{u}-t^{v}-y$ does not admit solutions in positive integers might have first appeared in a Russian compilation of problems of olympiad caliber...

What other notable variations of (*) do you know?

Please let me thank you in advance for your learned replies.

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