$\det(M)= 1$. Since we choose the Killing vector fields $\xi_\phi$ and $\xi_\psi$ to be spacelike, all the eigenvalues of $M$ are real and positive. Therefore, there exists an $G_{2(2)}$ matrix $\hat g$ such that $$\begin{aligned}
  M = \hat g \hat g^T \,.\end{aligned}$$ We define a current matrix as $$\begin{aligned}
 J_i = M^{-1} \partial_i M \,,\end{aligned}$$ which is conserved if the scalar fields are the solutions of the equation of motion derived by the action (\[eq:action\]). Then, the action (\[eq:action\]) can be written in terms of $J$ and $M$ as follows $$\begin{aligned}
S&=&\frac{1}{4}\int d\rho dz \rho {\rm tr}(J_iJ^i) \nonumber \\
 &=&\frac{1}{4}\int d\rho dz 
    \rho {\rm tr}(M^{-1}\partial_iMM^{-1}\partial^iM)\,. \end{aligned}$$ Thus, the matrix $M$ completely specify the solutions to our system.

Let us now consider two sets of field configurations, $M_{[0]}$ and $M_{[1]}$, that satisfy the equations of motion derived from the action, eq. (\[eq:action\]). We denote the difference between the value of the functional obtained from the field configuration $M_{[1]}$ and the value obtained from $M_{[0]}$ as a bull’s eye $\stackrel{\odot}{}$, e.g., $$\begin{aligned}
\stackrel{\odot}J{}^i=J^i_{[1]}-J^i_{[0]} \,,\end{aligned}$$ where the subscripts ${}_{[0]}$ and ${}_{[1]}$ denote, respectively, the quantities associated with the field configurations $M_{[0]}$ and $M_{[1]}$. The deviation matrix, $\Psi$, is then defined by $$\begin{aligned}
\Psi=\stackrel{\odot}MM^{-1}_{[0]}=M_{[1]}M^{-1}_{[0]}-{\bf 