  A theory of partial separability for classical Hamiltonian systems is
proposed in the context of Haantjes geometry.
  As a general result, we show that the knowledge of a non-semisimple
symplectic-Haantjes manifold for a given Hamiltonian system is sufficient to
construct sets of coordinates (called Darboux-Haantjes coordinates) which allow
both the partial separability of the associated Hamilton-Jacobi equations and
the block-diagonalization of the operators of the corresponding Haantjes
algebra.
  We also introduce a novel class of Hamiltonian systems, characterized by the
existence of a generalized St\"ackel matrix, which by construction are
partially separable. They widely generalize the known families of partially
separable Hamiltonian systems. Our systems can be described in terms of
semisimple but non-maximal-rank symplectic-Haantjes manifolds.
