Unequal-hemisphere dynamo rotating wave ($|B_{1,2}{|}^2=(a\pm \sqrt{a^2-4{ϵ}_1^2})/(2p)$)

For nonzero coupling ${ϵ}_1$, an alpha-quenching rotating wave can be made real in its toroidal amplitudes by a common phase rotation. Set $a={\alpha }_0-2\omega /\Omega$. The amplitude equations are $(a-p{B}_1^2)B_1+{ϵ}_1{B}_2=0$ and $(a-p{B}_2^2)B_2+{ϵ}_1{B}_1=0$. Subtracting after cross multiplication gives $(B_1^2-B_2^2)(p{B}_1{B}_2+{ϵ}_1)=0$. Unequal magnitudes therefore require $B_1{B}_2=-{ϵ}_1/p$ and $B_1^2+B_2^2=a/p$. They exist exactly when $a>2|{ϵ}_1|$ and have the displayed squared amplitudes. Exchanging hemispheres yields the companion solution. This breaks equatorial reflection symmetry and combines both dipole and quadrupole parity in a mean-field dynamo. Existence does not alone prove nonlinear stability.