= Solution
Suppose $\zeta>0$ and all $m_i$ are distinct. A nonzero $\phi_j$ forces $x=m_j$. Distinctness then makes every other charged field vanish, and $\sum_i\widetilde\phi_i\phi_i=0$ forces $\widetilde\phi_j=0$. The <D-flatness> equation fixes $|\phi_j|^2=\zeta$, while its phase is removed by the gauge group. There is therefore one isolated <supersymmetric vacuum> for each flavor,
$$
\boxed{x=m_j,\quad |\phi_j|^2=\zeta,\quad
\widetilde\phi_j=0,\qquad j=1,\ldots,N},
$$
for a total of $N$ vacua. Negative $\zeta$ gives the analogous vacua with $\widetilde\phi_j$ nonzero.
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