  We use Clifford algebras to construct a unified formalism for studying
constant extrinsic curvature immersed surfaces in riemannian and
semi-riemannian $3$-manifolds in terms of immersed bilegendrian surfaces in
their unitary bundles. As an application, we provide full classifications of
both complete and compact immersed bilegendrian surfaces in the unit tangent
bundle $\text{U}\Bbb{S}^3$ of the $3$-sphere.
