For $x_i \in \mathbb{Z}$, let $\{x_i\}$ be a fundamental solution to the equations: $$ \sum_{i= 1}^N x_i = \sum_{i=1}^N x_i^3 = 0 $$ if $x_i \in \{x_i\} \Rightarrow -x_i \notin \{x_i\}$.

For instance, a fundamental solution with $N=7$ is given by

$$ x_1 = 4, \quad x_2 = x_3 = x_4 = -3, \quad x_5 = x_6 = 2, \quad x_7 = 1 $$

What is the minimum $N$ for which a fundamental solution exists?