First let me put the question succinctly:
For whole numbers $m$ and $r$ how many integer pairs $(x,y)$ satisfy the equation $|x|^m+|y|^m=r$?
Now for exposition:
For some motivations to this questions check out: Computing $\beta(\frac{1}{m},\frac{1}{m})$
For a $m=2$ the result can be given in general which I will do below but the sake of example consider the case $m=2$ and $r=45$. Then there are $8$ integer pairs (x,y) such that $x^2+y^2=45$ and this can be seen in the picture below:
Generally speaking for the number of integer pairs that satisfy $x^2+y^2=r$
is equal to $$4\sum_{d|r} \chi_2(r)$$ where $$\chi_2(x) =
\begin{cases}
-1 & \quad \text{if } x \text{ is congruent to 3 mod 4}\\
1 & \quad \text{if } x \text{ is congruent to 1 mod 4}\\
0 & \quad \text{if } x \text{ is congruent to 0 mod 2}
\end{cases}$$
Now let's look at a case for $m\neq 2$. Consider the case $m=4$ and $r=97$. There are $8$ integer pairs that satisfy the equation $x^4+y^4=97$ as seen in the diagram below.
Is there a way to know in general how many solutions there are to $|x|^m+|y|^m=r$?