  Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian
manifold $(M,g)$, we study generic properties of unitary connections
$\nabla^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we
show that twisted Conformal Killing Tensors (CKTs) are generically trivial when
$\dim(M) \geq 3$, answering an open question of
Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the
existence of twisted CKTs is the only obstruction to solving exactly the
twisted cohomological equations which may appear in various geometric problems
such as the study of transparent connections. The main result of this paper
says that these equations can be generically solved. As a by-product, we also
obtain that the induced connection $\nabla^{\mathrm{End}(\mathcal{E})}$ on the
endomorphism bundle $\mathrm{End}(\mathcal{E})$ has generically trivial CKTs as
long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually,
we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the
geodesic flow is Anosov on the unit tangent bundle), the connections are
generically $\textit{opaque}$, namely there are no non-trivial subbundles of
$\mathcal{E}$ which are generically preserved by parallel transport along
geodesics. The proofs rely on the introduction of a new microlocal property for
(pseudo)differential operators called $\textit{operators of uniform divergence
type}$, and on perturbative arguments from spectral theory (especially on the
theory of Pollicott-Ruelle resonances in the Anosov case).
