  Consider the general complex polynomial external field $$
V(z)=\frac{z^{k}}{k}+\sum_{j=1}^{k-1} \frac{t_j z^j}{j}, \qquad t_j \in
\mathbb{C}, \quad k \in \mathbb{N}. $$ Fix an equivalence class $\mathcal{T}$
of admissible contours whose members approach $\infty$ in two different
directions and consider the associated max-min energy problem. When $k=2p$, $p
\in \mathbb{N}$, and $\mathcal{T}$ contains the real axis, we show that the set
of parameters $t_1, \cdots, t_{2p-1}$ which gives rise to a regular $q$-cut
max-min (equilibrium) measure, $1 \leq q \leq 2p-1 $, is an open set in
$\mathbb{C}^{2p-1}$. We use the implicit function theorem to prove that the
endpoint equations are solvable in a small enough neighborhood of a regular
$q$-cut point. We also establish the real-analyticity of the real and imaginary
parts of the end-points for all $q$-cut regimes, $1 \leq q \leq 2p-1$, with
respect to the real and imaginary parts of the complex parameters in the
external field. Our choice of even $k$ and the equivalence class $\mathcal{T}
\ni \mathbb{R}$ of admissible contours is only for the simplicity of exposition
and our proof extends to all possible choices in an analogous way.
