  We provide results for the leading superconducting instabilities for a model
pertaining to Sr$_2$RuO$_4$ obtained within spin-fluctuation mediated
superconductivity in the very weak-coupling limit. The theory incorporates
spin-orbit coupling (SOC) effects both in the band structure and in the pairing
kernel in the form of associated magnetic anisotropies. The leading
superconducting phase is found to be $d_{x^2-y^2}$ and a nodal $s$-wave state.
However, the odd-parity helical solution can become leading either for small
SOC and Hund's coupling $J$ in the weak $U$-limit, or in the opposite limit
with large SOC and $J$ at larger values of the Hubbard-$U$. The odd-parity
chiral solution is never found to be leading. Finally we discuss the form of
the resulting superconducting spectral gaps in the different explored parameter
regimes.
