We theoretically study the symmetry properties of the single-band Hubbard model with general spin-orbit coupling (SOC) on the kagome lattice. We show that the global U(1) spin-rotational symmetry is present in the Hubbard Hamiltonian owing to the inversion symmetry centered at the sites. The corresponding spin Hamiltonian has, therefore, SO(2) spin-rotational symmetry, which can be captured by including SOC nonperturbatively. The exact classical ground states, which we obtain for arbitrary SOC, are governed by the SU(2) fluxes associated with SOC threading the constituent triangles. The ground states break the SO(2) symmetry, and the associated Berezinsky-Kosterlitz-Thouless transition temperature is determined by the SU(2) fluxes through the triangles, which we confirm by a finite temperature classical Monte Carlo simulation.