SO(5) superconductors in a Zeeman magnetic field

The generic symmetry of a system under a uniform Zeeman magnetic field is $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1).$ However, we show that $\mathrm{SO}(5)$ models in the presence of a finite chemical potential and a finite Zeeman magnetic field can have an exact $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry. This principle can be used to test $\mathrm{SO}(5)$ symmetry at any doping level.