It is predicted that a new class of systems -superconductor/normal metal (S/N) heterostructures can reveal the in-plane Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) instability under nonequilibrium conditions at temperatures close to the critical temperature. It does not require any Zeeman interaction in the system. For S/N heterostructures under non-equilibrium distribution there is a natural easily adjustable parameter -voltage, which can control the FFLO-state. This FFLO-state can be of different types: plane wave, stationary wave and, even, 2D-structures are possible. Some types of the FFLO-state are accompanied by the magnetic flux, which can be observed experimentally. All the types of the FFLO-state can be revealed through the temperature dependence of the linear response of the system on the applied magnetic field near Tc, which strongly differs from that one for the homogeneous state.PACS numbers: 74.45.+c, 74.40.Gh There are two mechanisms of superconductivity destruction by a magnetic field: orbital effect and the Zeeman interaction of electron spins with the magnetic field. Usually the orbital effect is more restrictive. However there are several classes of systems, where the orbital effect is strongly weakened (systems with large effective mass of electrons 1,2 , thin films and layered superconductors under in-plane magnetic field 3 ) or even completely absent (superconductor/ferromagnet (S/F) heterostructures 4,5 ). Then the Zeeman interactions of electron spins with a magnetic or an exchange field is responsible for the superconductivity destruction.The behavior of a superconductor with a homogeneous exchange field h was studied long ago 6-9 . It was found that homogeneous superconducting state becomes energetically unfavorable above the paramagnetic (Pauli) limit h = ∆ 0 / √ 2, where ∆ 0 is the zero-temperature superconducting gap. As it was predicted by Larkin and Ovchinnikov 6 and by Fulde and Ferrell 7 , in a narrow region of exchange fields exceeding this value superconductivity can appear as an inhomogeneous state with a spatially modulated Cooper pair wave function (FFLOstate).Now there is a growing body of experimental evidence for the FFLO phase, generated by the applied magnetic field, reported from various measurements 10-24 . However, any unambiguous experimental results, which can be interpreted only as a fingerprint of the FFLO-state, are not reported by now.On the other hand, it has been predicted recently 25 that the FFLO-state can be realized in S/F heterostructures, where S is a singlet s-wave superconductor. Here we mean the so-called in-plane FFLO-state, where the superconducting order parameter profile is modulated along the layers. It should be distinguished from the normal to the S/F interface oscillations of the condensate wave function in the ferromagnetic layer, which are well investigated as theoretically, so as experimentally 4,5,26 .In this paper we show that the in-plane FFLO-state can be the most energetically favorable state in S/N heterostructures under the non-equili...