dg.differential geometry – Reference for non-parallel harmonic $k$-forms

I want to get some deep understanding on closed orientable Riemannian manifolds admitting $k$-forms ($kgeq 2$) $omega$ that satisfices the following conditions:
$$nabla omeganeq 0,quad Deltaomega=0.$$
where $Delta=(delta +d)^2$ is the Laplace-Beltrami operator (Hodge Laplacian). What other useful properties can these forms have? Any reference would be greatly appreciated.