Seeing as someone asked, this is my solution.

${x}^{2}+{y}^{2} = 849 - {x}^{2}{y}^{2}$

${x}^{2}{y}^{2} + {x}^{2}+{y}^{2} = 849$

${x}^{2}{y}^{2} + {x}^{2}+{y}^{2} + 1 = 850$

$({x}^{2}+1)({y}^{2}+1)=850$

The pairs of factors of 850 can now be considered.

$850= 1\times 850$

$850= 2\times 425$

$850= 5\times 170$

$850= 10\times 85$

$850= 17\times 50$

$850= 25\times 34$

It is clear that there are two occasions when both in the pair are one more than a square number so the possible solutions are:

${x}^{2}+1=5$ and ${y}^{2}+1=170$ (and vice versa)

or ${x}^{2}+1=17$ and ${y}^{2}+1=50$ (and vice versa)

Therefore $x=2$ or $13$ or $4$ or $7$ .

Of course, I had the advantage that I wrote the question for a friend with certain techniques in mind, though it is great to see other methods that I hadn't considered!

$x^2+y^2=849-x^2y^2$

$\Rightarrow x^2+y^2+x^2y^2=849 \\ \quad x^2y^2 + x^2+y^2+ 1=850 \\ \quad (x^2+1)(y^2+1) = 850$

We can use a spreadsheet to do the computation. Mine is as follows:

We note that $x=\{ 2,4,7,13\}$ . Therefore, the required answer is $\boxed{24713}$

In the first quadrant, the maximum values of $x$ and $y$ are both less than 30.

It's a simple matter to check the space of integer values of $x$ and $y$ for those that adhere to this relation.

How did the author of the problem do it?

PS: Incidentally, this way of providing for solutions of problems opens up a lot of possibilities!