Bi-presymplectic chains of 1-forms of co-rank 1 are considered. The conditions under which such chains represent some Liouville integrable systems and the conditions under which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of a dual Poisson-presymplectic pair is used, and the concept of d-compatibility of Poisson bivectors and d-compatibility of presymplectic forms is introduced. It is shown that bi-presymplectic representation of a related flow leads directly to the construction of separation coordinates in a purely algorithmic way. As an illustration, bi-presymplectic and bi-Hamiltonian chains in are considered in detail. © 2009 IOP Publishing Ltd.