Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and $$ \mathrm{Ricc}(g)=\lambda g, $$ $h$ a symmetric $2$covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well as the rough or connection Laplacian on symmetric $2$covariant tensors, and $\delta_g$ the divergence on symmetric $2$covariant tensors as well as on $1$forms, so $\delta_g\delta_gh$ is the doubledivergence and is a function. Write $\mathrm{tr}h$ for the metric trace of $h$ wrt $g$. Consider the following PDE: $$ \Delta(\mathrm{tr}h)=\lambda\mathrm{tr}h+\frac{m2}{2}\delta_g\delta_gh. $$ Questions: Does this PDE have only the trivial solution $h=0$? Are there conditions under which it has only the trivial solution? I'm interested in the case that $\lambda<0$.

5$\begingroup$ Can you give some context ? You have only one scalar PDE to prescribe $m(m+1)/2$ functions, so it seems a bit strange to conclude that $h$ is zero unless you have some example/evidence/extra information in mind... $\endgroup$ – Thomas Richard Oct 2 '15 at 7:15
There are lots of nontrivial solutions for any negative $\lambda$. Here's how to construct them all.
First, we can decompose an arbitrary symmetric $2$tensor $h$ as $h=fg+u$, where $f$ is a scalar function and $u$ is tracefree. It follows that $\operatorname{tr} h = mf$, and $\delta_g\delta_g h = \Delta f + \delta_g\delta_g u$. After some simplification, therefore, your equation becomes $$ \left(\Delta  \frac{2m\lambda }{3m  2}\right) f = \frac{m2}{3m2} \delta_g\delta_g u.\tag{$*$} $$ Since the eigenvalues of $\Delta$ are all nonnegative, the fact that $\lambda<0$ ensures that the operator in parentheses on the lefthand side is invertible on $C^\infty(M)$. Thus we can choose $u$ to be an arbitrary tracefree symmetric $2$tensor field, and then let $f$ be the unique solution to ($*$).
Notice that this has nothing to do with the manifold being Einstein. This argument works on any compact Riemannian manifold for any negative constant $\lambda$.
The case $\lambda\ge 0$ is a bit more complicated. In that case, if $2m\lambda /(3m2)$ isn't an eigenvalue of $\Delta$ the same argument works; but if it happens to be an eigenvalue, then there will be solutions only for those $u$ for which $\delta_g\delta_g u$ is $L^2$orthogonal to the kernel of the operator on the lefthand side.