**Setting**

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear operator acting on vector fields $L:\mathfrak{X}(M)\rightarrow \mathfrak{X}(M)$, 

$$
Lu=-\nabla ^*\nabla u+\mathrm{grad}\,\mathrm{div}\,u+\mathrm{Ric}\,u
$$
where, in a local orthonormal frame $\nabla ^*\nabla u=-\sum_{i}\nabla_{E_{i},E_{i}}^{2}u$; the operators $\mathrm{grad}:C^{\infty}(M)\rightarrow\mathfrak{X}(M)$ and $ \mathrm{div}:\mathfrak{X}(M)\rightarrow C^{\infty}(M)$ are the usual Riemannian versions of the operators from vector calculus; and $\mathrm{Ric}:\mathfrak{X}(M)\rightarrow \mathfrak{X}(M)$ is the Ricci operator, given by $\mathrm{Ric}\,u=\sum_{i}R(u,E_{i})E_{i}$, with $R$ being the $(3,1)$-curvature tensor. 

The operator $L$ has the following geometric significance: one can write $L=2\,\mathrm{Def}^{*}\mathrm{Def}$, where $\mathrm{Def}:\mathfrak{X}(M)\rightarrow S^{2}T^{*}M$ is the deformation operator and $\mathrm{Def}^{*}$ is its formal adjoint (which coincides with the tensor divergence operator). The former is given by either,
$$
\mathrm{Def}\, u=\frac{1}{2}\mathcal{L}_{u}g=\frac{1}{2}\nabla u+\frac{1}{2}(\nabla u)^{T}
$$
where $\mathcal{L}_{u}g$ is the Lie derivative of the Riemannian metric by $u$, and here
$\nabla u$ is viewed as a covariant tensor field of type $(2,0)$. 

**The boundary value problem**

I am interested in solutions to $Lu=0$. When $M$ is a compact manifold without boundary,  $Lu=\mathrm{Def}^{*}\mathrm{Def}u=0$ must imply that $\mathcal{L}_{u}g=0$, which is to say, $u$ is a Killing field.

In a compact Riemannian manifold with boundary, however, the equation $Lu=0$ has to be supplemented with boundary conditions. Denoting by $N$ as the normal to the boundary, I am interested in the boundary prescription:

$$
\mathcal{L}_{u}g(N,\cdot)=g(N,\cdot) \,\,\, \text{on}\, \partial M\,\,\,(*).
$$

This condition can be viewed as generalized Neumann condition, and it can be verified that together with $Lu=0$ the resulting $BVP$ is elliptic. By standard theory of elliptic BVPs for self adjoint operators, it can be shown that $Lu=0$ supplemented with $(*)$ always yields a unique solution, modulo Killing fields, regardless of the background metric $g$. In particular, this solution cannot be a Killing field (otherwise, both sides of $(*)$ would vanish). 

**My questions**

1) The Lie derivative of any metric construction is invariant by Killing fields: this includes the Levi-civita connection and any kind of curvature. Although the unique solution to the BVP $Lu=0$ + (*) cannot be a Killing field, is there any hope that it preserves such geometric quantities? For example, the connection, the curvature tensor, or any other metric consturction? 

2)  If $(M,g)$ is an Euclidean domain with smooth boundary, then by the uniqueness clause

$$
u=\mathrm{grad}\,r^{2}\Longrightarrow \mathcal{L}_{u}g=\frac{1}{2}\mathrm{Hess}\,r^{2}=\sum_{i} dx_{i}^{2}=g
$$
  where $r^{2}=\sum_{i}x_{i}^{2}$ is the distance function from the 
  origin. The equation $g=\mathcal{L}_{u}g$ in general doesn't have a 
  solution (not all Riemannian manifolds admit homotheties). But in 
  general, are there other cases in which $u$ can be represented as the 
  gradient of a scalar function other than this case?

3) Any other geometric facts you might know are most welcome. 

Many thanks!