Nonvertical qualification in the steady-top double-root condition ($P(u_0)=P^{\prime }(u_0)=0,-1<u_0<1$)

After fixing axial and vertical angular momenta, the heavy symmetric top has energy $A{\dot{\theta }}^2/2+V_{\mathrm{eff}}(\cos \theta )$. A fixed angle strictly between zero and $\pi$ requires both $E=V_{\mathrm{eff}}(u_0)$ and $V_{\mathrm{eff}}^{\prime }(u_0)=0$. Thus $P(u)=2A(1-u^2)(E-V_{\mathrm{eff}}(u))$ has a repeated interior root. A vertical axis belongs to the excluded endpoints: the upright stationary top has $P(u)=2Amga(1-u{)}^2(1+u)$, whose only repeated root is one. An assertion of an interior common root therefore needs the nonvertical hypothesis.