  We characterize biharmonic anti-invariant surfaces in $3$-dimensional
generalized $(\kappa, \mu)$-manifolds with non-zero constant mean curvature by
means of the scalar curvature of the ambient space and the mean curvature. In
addition, we give a method for constructing infinity many examples of
biharmonic submanifolds in a certain $3$-dimensional generalized $(\kappa,
\mu)$-manifold. Moreover, we determine $3$-dimensional generalized $(\kappa,
\mu)$-manifolds which admit a certain kind of proper biharmonic foliation.
