extremely confusing integer question

Question

The reverse of a $2$-digit integer is the integer obtained by reversing the order of the $2$ digits. For example the reverse of $43$ is $34$. How many $2$-digit positive integers $N$ exist with the property that the sum of $N$ and the reverse of $N$ is the square of an integer?

Answer 1

Hint: let the integer be $\overline{ab}=10a+b$ where $1 \le a,b \le 9$ are its decimal digits (assuming $a,b \ne 0$ so that both numbers have, in fact, two digits). Then: $$\overline{ab}+\overline{ba}= 10a+b+10b+a=11(a+b)$$ Now, for $11(a+b)$ to be a perfect square, $a+b$ must be a multiple of $11\,$, so $\cdots$