We measure the gate voltage ($V_g$) dependence of the superconducting properties and the spin-orbit interaction in the (111)-oriented LaAlO$_3$/SrTiO$_3$ interface. Superconductivity is observed in a dome-shaped region in the carrier density-temperature phase diagram with the maxima of superconducting transition temperature $T_c$ and the upper critical fields lying at the same $V_g$. The spin-orbit interaction determined from the superconducting parameters and confirmed by weak-antilocalization measurements follows the same gate voltage dependence as $T_c$. The correlation between the superconductivity and spin-orbit interaction as well as the enhancement of the parallel upper critical field, well beyond the Chandrasekhar-Clogston limit suggest that superconductivity and the spin-orbit interaction are linked in a nontrivial fashion. We propose possible scenarios to explain this unconventional behavior.
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