Suppose that $\omega$ is an 1-form on a Reimannain Manifold $(M,g)$ and $s$ is a $(0,2)$ symmetric tensor which be considered as $(1,1)$ symmetric tensor whenever it is convenient. If for all $s$ the following integral vanishes,

$$\int_M <\big( -\mathrm{Ric} +\dfrac{1}{2}Rg+2\mathrm{Sym}(\nabla \omega) +\omega\otimes \omega -|\pi |^2 g\big) ,s>dV_g\\ +2\int_M (<s,\nabla U>+<\vec{\nabla}\mathrm{tr}(s),U> )dV_g =0$$

Then what we can obtain from this equality?

Note that $U$ is is defined by $w=g(U,.)$.