  We consider the spin and charge Josephson current between two non-uniform
Fulde-Ferrel-Larkin-Ovchinnikov superconductors with helimagnetic order. We
demonstrate that the presence of the helimagnetic phase generates a spin
Josephson effect and leads to additional contributions to both single-particle
and Josephson charge current. It is shown that for such systems the AC effect
differs more radically from the DC effect than in the case of a BCS
superconductor with helimagnetic order considered earlier in the literature [M.
L. Kuli\'c and I. M. Kuli\'c, Phys. Rev. B {\bf 63}, 104503 (2001)] where a
spin Josephson current has also been found. In our system the most interesting
effect occurs in the presence of an external magnetic field and in absence of
voltage, where we show that the charge Josephson current can be tuned to zero
while the spin Josephson current is non-vanishing. This provides a well
controlled mechanism to generate a spin supercurrent in absence of charge
currents.
