linear algebra - Recovering symmetry in coupled oscillators

Consider a pair of LC oscillators, one with capacitance $C_1$ and inductance $L_1$ and the other with capacitance $C_2$ and inductance $L_2$. Suppose they're connected through a capacitor $C_g$. We want to find the normal modes and frequencies.

If we write out Kirchhoff's laws, we find \begin{align} V_1 + \ddot{V}_1 \left(1 + \epsilon_1 \right)/\omega_1^2 - (\epsilon_1/\omega_1^2)\ddot{V}_2 &= 0 \\ V_2 + \ddot{V}_2 \left(1 + \epsilon_2 \right)/\omega_2^2 - (\epsilon_2/\omega_2^2)\ddot{V}_1 &= 0 \\ \end{align} where $\epsilon_i \equiv C_g / C_i$ and $\omega_i^2 \equiv 1/L_i C_i$. These equations can be written in matrix form as $$ \left( \begin{array}{c} V_1 \\ V_2 \end{array} \right) = \left( \begin{array}{cc} (1 + \epsilon_1)/\omega_1^2 & - \epsilon_1 / \omega_1^2 \\ - \epsilon_2 / \omega_2^2 & (1 + \epsilon_2)/\omega_2^2 \\ \end{array} \right) \left( \begin{array}{c} \ddot{V}_1 \\ \ddot{V}_2 \end{array} \right) \tag{$\star$} \, . $$ Now if $L_1 = L_2$ and $C_1 = C_2$ then $\epsilon_1 = \epsilon_2 \equiv \epsilon$ and $\omega_1 = \omega_2 \equiv \omega_0$ and the matrix equation becomes $$ \left( \begin{array}{c} V_1 \\ V_2 \end{array} \right) = \left( \begin{array}{cc} (1 + \epsilon)/\omega_0^2 & - \epsilon / \omega_0^2 \\ - \epsilon / \omega_0^2 & (1 + \epsilon)/\omega_0^2 \\ \end{array} \right) \left( \begin{array}{c} \ddot{V}_1 \\ \ddot{V}_2 \end{array} \right) \, . $$ In this particular case, the matrix can be written in the nice form $$ \frac{1 + \epsilon}{\omega_0^2} \, \mathbb{I} - \frac{\epsilon}{\omega_0^2} \sigma_x \tag{$\star \star$} $$ and it's pretty easy to find the normal modes and normal frequencies.$^{[a]}$

However, when the oscillators aren't identical, e.g. Eq. ($\star$), expressions for the normal modes and frequencies are pretty messy. Is there a transformation we can apply to ($\star$) to bring it into a simple form like ($\star \star$) so that the mode analysis results in simpler equations?

Perhaps another way to ask this would be to ask for a systematic way to rescale the variables so that the matrix in the equations of motion is symmetric or perhaps Hermitian.

[a] The frequencies are $\omega_0$ (even mode) and $\omega_0 / \sqrt{1 + 2 \epsilon}$ (odd mode).