  We study the effect of the coupling between the electronic ground state of
high spin alkaline-earth fermionic atoms and their metastable optically excited
state, when the system is confined in a one-dimensional chain, and show that
the system provides a possible realization of a finite momentum pairing
(Fulde-Ferrell-Larkin-Ovchinnikov-like) state without spin- or bare mass
imbalance. We determine the $\beta$-functions of the renormalization group
trajectories for general spin and analyze the structure of the possible gapped
and gapless states in the hydrodynamic limit. Due to the SU(N) symmetry in the
spin space, complete mode separation can not be observed even in the fully
gapless 2N-component Luttinger liquid state. Contrary, 4 velocities
characterize the system. We solve the renormalization group equations for
spin-9/2 strontium-87 isotope and analyze in detail its phase diagram. The
fully gapless Luttinger liquid state does not stabilize in the two-orbital
system of the $^{87}$Sr atoms, instead, different gapped non-Gaussian fixed
points are identified either with dominant density or superconducting
fluctuations. The superconducting states are stable in a nontrivial shaped
region in the parameter space as a consequence of the coupling between the two
electronic states.
