$$\delta^{(2)}S+S^{(GF)}= -\frac{v}{e}\int_{\mathcal{R}^{2,1}} \, dx^0 dx^1 dx^2\,\left\{ \xi^{\rm t}(x^0,\vec{x})\left[\frac{\partial^2}{\partial (x^0)^2}+{\cal H}^+\right] \xi(x^0,\vec{x})\right\}+o(\xi^3)$$ where $${\cal H}^+= \left( \begin{array}{cccc}
-\Delta + |\psi|^2 & 0 & -2D_1 \psi_2 & 2 D_1 \psi_1 \\
0 & -\Delta +|\psi|^2 & -2 D_2 \psi_2 & 2 D_2 \psi_1 \\
-2 D_1 \psi_2 & -2 D_2\psi_2 & -\Delta +\frac{1}{2} (3|\psi|^2-1)+V_kV_k & -2 V_k \partial_k -\partial_k V_k \\
2D_1\psi_1 & 2 D_2 \psi_1 & 2V_k \partial_k + \partial_k V_k & -\Delta +\frac{1}{2} (3|\psi|^2-1) + V_kV_k
 \end{array} \right) \label{operator1}$$ is the Hessian or second-order fluctuation operator and terms of third and quartic order in the perturbations are neglected. In the operator ${\cal H}^+$ we denote: $D_1\psi_1 = \partial_1 \psi_1 + V_1 \psi_2$, $D_2\psi_1 = \partial_2 \psi_1 + V_2 \psi_2$, $D_1\psi_2 = \partial_1 \psi_2 - V_1 \psi_1$ and $D_2\psi_2 = \partial_2 \psi_2 - V_2 \psi_1$. In this background gauge the classical energy up to the quadratic order in the small fluctuations is now easily derived: $$H^{(2)}+H^{(GF)}=\frac{v^2}{2}\int_{\mathbb{R}^2}\, dx^1dx^2 \, \left\{\frac{\partial\xi^t}{\partial t}(\vec{x},t)\frac{\partial\xi}{\partial t}(\vec{x},t)+\xi^t(\vec{x},t){\cal H}^+\xi(\vec{x},t)\right\} +o(\xi^3)\, .$$ We impose finiteness of the norm on the static fluctuations, equivalently the fixed time perturbations, $\xi(\vec{x})$: $\|\xi(\vec{x})\|^2  = \int_{\mathbb{R}^2} d^2x [ (a_1(\vec{x}))^2 + (a_2(\vec{x}))^2 + (\varphi_1(\vec{x}))^2 + (\varphi_2(\vec{x}))^2 ] < +\infty$. Thus, the four component vectors of 