I have the following system of first order quasi-linear pde:

$$ -(\Delta+1) a^{\alpha\beta} [b_{\beta\rho} I_{\alpha;\sigma}+b_{\beta\sigma} I_{\alpha;\rho}] + a^{\alpha\beta} [(\Delta+1) b_{\beta\rho;\sigma}+b_{\beta\rho}\Delta_{;\sigma}+b_{\beta\sigma}\Delta_{;\rho}]I_\alpha +a^{\alpha\beta} b_{\rho\sigma}I_\alpha I_\beta=0. \,\,\,\,\,\,\,(1)$$

Here, the subscript $(\cdot)_{;}$ denotes the covariant derivative with respect to the usual surface Christoffel symbols $\Gamma^\mu_{\alpha\beta}\equiv\frac{1}{2}a^{\mu\nu}(a_{\nu\alpha,\beta}+a_{\nu\beta,\alpha}-a_{\alpha\beta,\nu})$ on a non-compact surface $S$, embedded in $\mathbb{R}^3$, with first fundamental form $a_{\alpha\beta}$ and second fundamental form $b_{\alpha\beta}$. $\Delta(\ne -1)$ is a sufficiently smooth known function defined on $S$.

It can be observed that $I_\alpha=0$ clearly satisfies the above pde, and in my research problem, I want only this solution to occur. But I have to argue about the uniqueness of the above solution. Otherwise, if there are other solutions, I have to rule them out on `physical' grounds.

I wonder if anything could be said about the uniqueness of this pde, namely conditions on the coefficients, and, if possible, existence.

**A brief background of the problem:**
On an embedded smooth enough non-compact surface $s$ in $\mathbb{R}^3$, with first and second fundamental form $A_{\alpha\beta}$ and $B_{\alpha\beta}$ respectively, given are five real fields: $[E_{\alpha\beta}]$ (a symmetric positive definite matrix), $[\Lambda_{\alpha\beta}]$ (a non-necessary symmetric invertible matrix), ${\Lambda_\alpha}$, $\Delta_\alpha$ and $\Delta (\ne -1)$. We are required to construct, out of this information, a surface $S$ given by $\mathbf{r}(\theta^\alpha)$, with a transverse vector field $\mathbf{d}(\theta^\alpha)$ attached to every point on $S$, and having first fundamental form $a_{\alpha\beta}=A_{\alpha\beta}+2E_{\alpha\beta}$ and a suitably constructed second fundamental form $b_{\alpha\beta}$ out of the given fields, so that the following are true:

$$\Lambda_{\alpha\beta}=\mathbf{d}_{,\beta}\cdot\mathbf{a}_\alpha+B_{\alpha\beta}$$ $$\Lambda_\alpha=\mathbf{d}_{,\alpha}\cdot\mathbf{n}$$ $$\Delta_\alpha=\mathbf{d}\cdot\mathbf{a}_{\alpha}$$ $$\Delta=\mathbf{d}\cdot\mathbf{n}-1$$

Here, $\mathbf{n}$ is the unit normal field on $S$ and $\mathbf{a}_\alpha=\mathbf{r}_{,\alpha}$.

The necessary and sufficient conditions for the above to happen are given by

$$\Lambda_{[\alpha\beta]}-\Delta_{[\alpha;\beta]}=0,$$ $$I_\alpha\equiv \Lambda_\alpha-\Delta_{,\alpha}+\Delta_{\beta}a^{\beta\gamma}b_{\gamma\alpha}=0,$$ and the Gauss and Codazzi-Mainardi relations of $a_{\alpha\beta}$ and $b_{\alpha\beta}$ where $b_{\alpha\beta}\equiv\dfrac{\Lambda_{\gamma\alpha}-B_{\gamma\alpha}-\Delta_{\gamma;\alpha}}{\Delta+1}$.

Here, the subscript $(\cdot)_{[.]}$ denotes the skew part of a matrix.

This has already been proved in *M. Epstein, A Note on Nonlinear Compatibility Equations for Sandwich Shells and Cosserat Surfaces, Acta Mechanica 31, 285-289 (1979)*. In order to put this result in a different light, I arrived at equation (1).