Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$ 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 $x \in \{x_i\} \Rightarrow -x \notin \{x_i\}$.
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?

We can get $N=6$ from $1+5+5=2+3+6$, $1^3+5^3+5^3=2^3+3^3+6^3$. 
We can get $N=5$ from $2+4+10=7+9$, $2^3+4^3+10^3=7^3+9^3$. 
