Quadratic even-odd mode interaction

A reflection acting as $(A,B)\mapsto (A,-B)$ forces an equivariant vector field to have $\dot{A}$ even and $\dot{B}$ odd in $B$. With the zero solution preserved, its quadratic normal form is $\dot{A}={\lambda }_{1}A+a_1{A}^2+a_2{B}^2$, $\dot{B}={\lambda }_{2}B+a_{3}AB$. For positive coefficients, every mixed equilibrium with $B\ne 0$ has Jacobian determinant $-2a_2{a}_3{B}^2<0$, hence is a saddle equilibrium. The two invariant-axis equilibria meet in a transcritical bifurcation, and the mixed branches meet them in pitchfork bifurcations.