According to the given information we obtain the Diophantine equation $(1) \;\; m^3 + n^3 = (m + n)^2$, i.e. $(m + n)(m^2 - mn + n^2) - (m + n)^2 = 0$, or equivalently $(m + n)(m^2 - mn + n^2 - m - n) = 0$, yielding $m = -n$ or $(2) \;\; m^2 - mn + n^2 - m - n = 0$. Equation (2) can be expressed as $m^2 - (n + 1)m + n^2 - n = 0$, yielding $m = \frac{n + 1 \pm d}{2}$, where the non-negative discriminant $d = (n + 1)^2 - 4(n^2 - n) = -3n^2 + 6n + 1 \geq 1$, which means $n(n - 2) \leq 0$, yielding $n \{0,1,2\}$. Inserting these three integers for $n$ in equation (2), we obtain the solutions $(m,n) = (0,0), (1,0), (0,1), (2,1), (1,2), (2,2)$, which together with $(m,n) = (t,-t) \: (|t| \in \mathbb{N})$ are all integer solutions of equation (1).