The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong–Wong type. Under certain conditions, these generalizations remain integrable. We further set up a gauge theory with a product gauge group, two complex scalar fields and a general charge matrix. The second species of vortices, when frozen, are interpreted as the magnetic impurity for all five vortex equations. We then give a geometric compatibility condition, which enables us to remove the constant term in all the equations. This is similar to the reduction from the Taubes equation to the Liouville equation. We further find a family of charge matrices that turn the five vortex equations into either the Toda equation or the Toda equation with the opposite sign. We find exact analytic solutions in all cases and the solution with the opposite sign appears to be new.

Manton 最近研究的五个可积涡旋方程被推广到包括 Tong-Wong 类型的磁性杂质。在某些条件下，这些概括仍然是可积分的。我们进一步建立了具有乘积规范群、两个复标量场和一般电荷矩阵的规范理论。当冻结时，第二种涡流被解释为所有五个涡流方程的磁性杂质。然后我们给出了一个几何兼容性条件，它使我们能够去除所有方程中的常数项。这类似于从 Taubes 方程到 Liouville 方程的简化。我们进一步找到了一组电荷矩阵，可以将五个涡旋方程转换为 Toda 方程或具有相反符号的 Toda 方程。