AbstractThe mixed Kaup-Newell (mKN) hierarchy, including the nonholonomic deformation of the KN equation, is obtained in the Lenard scheme. By the nonlinearisation of the Lax pair, the mKN hierarchy is reduced to a family of mixed, finite-dimensional Hamiltonian systems (FDHSs) that separate its temporal and spatial variables. It turns out that the Bargmann map not only gives rise to the finite parametric solutions of the mKN hierarchy but also specifies a finite-dimensional, invariant subspace for the mKN flows. The Abel-Jacobi variables are selected to linearise the mKN flows on the Jacobi variety of a Riemann surface, from which some quasi-periodic solutions of mKN hierarchy are presented by using the Riemann-Jacobi inversion.