  We propose an exact solution of a model describing a low energy behavior of
cold isospin S=3/2 fermionic atoms on a one-dimensional optical lattice.
Depending on the band filling the effective field theory has a form of a
deformed Gross-Neveu model with either $O(7)\times \mathbb{Z}_2$ (half filling)
or $U(1)\times O(5)\times \mathbb{Z}_2$ symmetry.
