assume that $\gamma$ is parameterized by conformal parameter. Keep in mind that $W_{\gamma}=0$ if and only if $$\label{EL1}
\dot{h}_1=3h_2\dot{h}_2,\quad D^2(S_{\gamma})=(h_2^2-2h_1)S_{\gamma}.$$ Let fix a second-order frame $A:I\to \mathrm{A}^{\uparrow}_+(2,n)$ and $$\mathbf{g}:(u,\tau)\in I\times (-\epsilon,\epsilon)\to \gamma_{\tau}(u)\in \mathcal{E}^{1,n-1}$$ be a compactly supported variation of $\gamma$ such that $\mathrm{supp}(\mathbf{g})\subseteq K$. Eventually shrinking the interval $(-\epsilon, \epsilon)$, we may assume that $\gamma_{\tau}$ is generic and equipped with its intrinsic orientation, for every $\tau$. Then, there is a differentiable map $$\mathcal{A}:(u,\tau)\in I\times (-\epsilon,\epsilon)\to \mathcal{A}_{\tau}(u)\in \mathrm{A}^{\uparrow}_+(2,n)$$ such that $\mathcal{A}_0=A$ and $\mathcal{A}_{\tau}$ is a second-order frame along $\gamma_{\tau}$, for every $\tau\in (-\epsilon,\epsilon)$. We then have $$\mathcal{A}^{-1}d\mathcal{A} = Qdu+\Lambda dt,$$ where $$Q=\left(
                                 \begin{array}{ccccc}
                                   0 & -m & v & 0 & 0 \\
                                   v & 0 & 0 & 0 & m \\
                                   0 & 0 & 0 & -^tp & 0 \\
                                   0 & 0 & p & \psi & 0 \\
                                   0 & v & 0 & 0 & 0 \\
                                 \end{array}
                               \right),$$ and $$\Lambda=\left(
                                 \begin{array}{ccccc}
                                   \lambda_0^0 & -\lambda_{n+1}^1 & \lambda^2_{n+1} & ^t\lambda_{n+1} & 0 \\
                                   \lambda^1_0 & 0 & \lambda^2_1 & ^t\lambda_1 & \lambda^1_{n+1} \\
                                   \lambda^2_0 & \lambda^2_1 & 0 & -^t\lambda_2 & \lambda^2_{n+1} \\
                                   \lambda_0 & \lambda_1 & \lambda_2 & \lambda & \lambda_{n+1} \\
                                   0 & -\lambda^1_0 