motion (\[eq: EOM\]) is invariant under rescaling $$A_M \to s A_M \ , \quad \xi \to s^2 \xi \ ,
\quad \zeta \to s^2 \zeta \ ,
\label{rescaleing_XYM}$$ we make this rescaling at each step of the algorithm in such a way that the second equation of is satisfied. The algorithm proceeds by updating $X_\mu$, $Y_a$ and $M$ with the increment $$\delta X_\mu = -\epsilon {\frac{{\partial}V}{{\partial}X^{*}_\mu}}\ ,\quad
\delta Y_a = -\epsilon {\frac{{\partial}V}{{\partial}Y^{*}_a}}\ ,\quad
\delta M = -\epsilon {\frac{{\partial}V}{{\partial}M^{*}}}\ ,
\label{updating_XYM}$$ where the stepsize $\epsilon$ is taken to be small enough to make sure that $V$ decreases at each step. We repeat (\[rescaleing\_XYM\]) and (\[updating\_XYM\]) until $V$ becomes as small as $O(10^{-4})$. The first equation of (\[eq:normalization\]) is not imposed on configurations during this procedure, and we use it only to determine the value of $\kappa$ after a classical solution is found. In this way, we can obtain infinitely many solutions with some $\kappa$ depending on the initial random configuration and the initial values of $\xi$ and $\zeta$.

typical solutions {#sec:typical-sol}
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In typical solutions generated by our algorithm, it turns out that the matrices $M$ and $Y_a$ take the form[^2] $$\label{eq: correspondence_MY}
M= \sqrt{\frac{N_{\rm Y}}{N_{\rm Y} - n_0 }}
\begin{pmatrix}
