Precise measurement of the branching fractions of $J/\psi \to \overline{\Lambda }{\pi }^{+}{\Sigma }^{-}+c.c.$ and $J/\psi \to \overline{\Lambda }{\pi }^{-}{\Sigma }^{+}+c.c.$

Based on a data sample of $\left(10087±44\right)×{10}^{6}\text{}\text{}J/\psi $ events collected with the BESIII detector, the branching fraction of $J/\psi \to \overline{\Lambda }{\pi }^{+}{\Sigma }^{-}+c.c.$ is measured to be $\left(1.221±0.002±0.038\right)×{10}^{-3}$, and the branching fraction of its isospin partner mode $J/\psi \to \overline{\Lambda }{\pi }^{-}{\Sigma }^{+}+c.c.$ is measured to be $\left(1.244±0.002±0.045\right)×{10}^{-3}$ with improved precision. Here the first uncertainties are statistical and the second ones systematic. The isospin symmetry of the $\Sigma $ baryon in charmonium hadronic decay and the "12% rule" are tested, and no violation is found. The potential of using these channels as $\Sigma $ baryon sources for nuclear physics research is studied, and the momentum and angular distributions of these sources are provided.