23. $x^{2}+2 x \sin x y+1=0$.

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23. $x^{2}+2 x \sin x y+1=0$.
23. Replace 1 with the sum $\sin ^{2} x y+\cos ^{2} x y$. The equation will take the form $(x+\sin x y)^{2}+\cos ^{2} x y=0$. Such an equality is possible only if $x+\sin x y=0$ and $\cos x y=0$. From this system, the values of $x$ and $y$ can easily be found. $x= \pm 1, y=-\frac{\pi}{2}+2 \pi k$.