  We formulate the theory for a diatomic molecule in a spatially degenerate
electronic state interacting with a non-resonant laser field and investigate
its rovibrational structure in the presence of the field. We report on
\textit{ab initio} calculations employing the double electron attachment
intermediate Hamiltonian Fock space coupled cluster method restricted to single
and double excitations for all electronic states of the Rb$_2$ molecule up to
$5s+5d$ dissociation limit of about 26.000$\,$cm$^{-1}$. In order to correctly
predict the spectroscopic behavior of Rb$_2$, we have also calculated the
electric transition dipole moments, non-adiabatic coupling and spin-orbit
coupling matrix elements, and static dipole polarizabilities, using the
multireference configuration interaction method. When a molecule is exposed to
strong non-resonant light, its rovibrational levels get hybridized. We study
the spectroscopic signatures of this effect for transitions between the
X$^1\Sigma_g^+$ electronic ground state and the A$^1\Sigma_u^+$ and b$^3\Pi_u$
excited state manifold. The latter is characterized by strong perturbations due
to the spin-orbit interaction. We find that for non-resonant field strengths of
the order $10^9$W/cm$^2$, the spin-orbit interaction and coupling to the
non-resonant field become comparable. The non-resonant field can then be used
to control the singlet-triplet character of a rovibrational level.
