  We study inverse boundary problems for first order perturbations of the
biharmonic operator on a conformally transversally anisotropic Riemannian
manifold of dimension $n \ge 3$. We show that a continuous first order
perturbation can be determined uniquely from the knowledge of the set of the
Cauchy data on the boundary of the manifold provided that the geodesic $X$-ray
transform on the transversal manifold is injective.
