An inverse problem for an inhomogeneous conformal Killing field equation

Let $g$ be a $C^{2,\alpha}$ Riemannian metric defined on a bounded domain $\Omega\subset R^2$ with $C^{3,\alpha}$ boundary and let $X$be a $C^{2,\alpha}$ vector field on $\bar{\Omega}$ satisfying $X\vert _{\partial\Omega}=0$. We show that if $l$ is a gradient field of a solution $u$ to the equation $\triangle_gu-\bigl\langle\nabla_{g\,}\sigma,\,\nabla_gu\bigr\rangle_g=0$ on $\Omega$, then both inner products $\bigl\langle l,X\bigr\rangle_g\,$ and $\bigl\langle l^\perp,X\bigr\rangle_g\,$ are uniquely determined by the restriction of the tensor ${\mathcal L}_X(g)-(e^\sigma\,\nabla_{g}\cdot(e^{-\sigma}X)) g$ to the gradient field $l$, where ${\mathcal L}_X(g)$ is the Lie derivative of the metric tensor $g$ under the vector field $X$ and $\sigma=log\sqrt{det(g)}$. This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.