  Bi-presymplectic chains of one-forms of arbitrary co-rank are considered. The
conditions in which such chains represent some Liouville integrable systems and
the conditions in which there exist related bi-Hamiltonian chains of vector
fields are presented. In order to derived the construction of bi-presymplectic
chains, the notions of dual Poisson-presymplectic pair, d-compatibility of
presymplectic forms and d-compatibility of Poisson bivectors is used. The
completely algorithmic construction of separation coordinates is demonstrated.
It is also proved that St\"{a}ckel separable systems have
bi-inverse-Hamiltonian representation, i.e. are represented by bi-presymplectic
chains of closed one-forms. The co-rank of related structures depends on the
explicit form of separation relations.
