has a closed-form solution as $$\label{equ:21}
\mathbf{t}=-\left( {{\mathbf{B}}^{T}}\mathbf{B} \right)^{-1}{{\mathbf{B}}^{T}}\mathbf{Au}$$ Substituting (\[equ:21\]) into (\[equ:20\]), we derive a cost function only involving $\rho, \alpha, \beta, \gamma$ as $$\label{equ:22}
{{C}_{\pi lp}}\left( \rho, \alpha, \beta, \gamma \right)={{\mathbf{u}}^{T}}{{\mathbf{C}}^{T}}\mathbf{Cu}={{\mathbf{u}}^{T}}\mathbf{Qu}$$ where ${\bf{C}}=  {{\bf{A}} - {\bf{B}}{{\left( {{{\bf{B}}^T}{\bf{B}}} \right)}^{ - 1}}{{\bf{B}}^T}{\bf{A}}} $. The elements in the vector $\mathbf{u}$ are second order monomials except for the constant term. Thus (\[equ:22\]) is a fourth order polynomial function. We compute the first order optimality conditions to get all the stationary points. This first-order condition contains four third order polynomials for $\rho ,\alpha ,\beta $ and $\gamma$ as $$\label{equ:23}
\begin{split}
{g_\rho } = \frac{{\partial {C_{\pi lp}}}}{{\partial \rho }} = 0, \quad {g_\alpha } = \frac{{\partial {C_{\pi lp}}}}{{\partial \alpha }} = 0, \\
{g_\beta } = \frac{{\partial {C_{\pi lp}}}}{{\partial \beta }} = 0, \quad {g_\gamma } = \frac{{\partial {C_{\pi lp}}}}{{\partial \gamma }} = 0.
\end{split}$$

According to Bézout’s theorem [@cox2006using], there exist at most ${{3}^{4}}=81$ solutions. We find that the polynomial system in (\[equ:23\]) only contains third and first degree monomials. This equation system stratifies the 2-fold symmetry [@larsson2016uncovering; @ask2012exploiting]. That is to say if any nontrivial $\mathbf{\xi}={{\left[ \rho ,\alpha ,\beta ,\gamma  \right]}^{T}}$ is a solution, $-\mathbf{\xi}$ is also a solution. Thus there are 