The given equation is equivalent to
$$(11x^5 - 31x) + (33y - 13y^5) = 2024.$$
The fifth power of any positive integer ends with the same digit as the number itself. Therefore, $11x^5$ ends with the same digit as $11x$, which in turn ends with the same digit as $x$. Since $31x$ also ends with the same digit, the difference $11x^5 - 31x$ ends with zero. Similarly, the difference $33y - 13y^5$ ends with zero. However, the sum of numbers ending in zero cannot end in four. Therefore, there are no integers that satisfy the given equation.