$E_{\textrm{tot}}$ defined by $$E_{\textrm{tot}} \equiv \frac{1}{2} I_{\textrm{eff}} \overline{n}(t)^2 + \overline{E}_{p}(t) + \overline E_{r}(t)$$ is conserved. It can be seen that there is another invariant, $$C_{I} \equiv \frac{1}{K_{p}}\ln \overline{E}_{p} + \frac{1}{K_{r}}\ln \overline{E}_{r},\label{eq:casimir}$$ which has been discussed in connection with a Casimir invariant [@MoffattTokieda2008; @Yoshida2016]. With these two conservatives, general solutions of the three-variable system (\[eq:diff-ne-1\])–(\[eq:diff-ne-3\]) should be periodic.

Let us consider the case where the spin is positive at $t=0$ and the sum of the oscillation energies are small compared to the spinning energy: $$\overline{n}(0) \equiv n_i >0, \quad 
\overline{E}_{p}(0) + \overline{E}_{r}(0) \ll \frac{1}{2} I_{\textrm{eff}} n_{i}^2.$$ For a typical rattleback, the pitching develops and the rolling decays as long as $\overline n>0$ as can be seen from Eqs. (\[eq:sign-typ-k\]), (\[eq:diff-ne-2\]) and (\[eq:diff-ne-3\]). Thus the rolling is irrelevant and can be ignored, i.e., $\overline E_r(t) = 0$, to estimate the time for reversal. Then we can derive the equation $$\frac{d\overline{n}(t)}{dt} = -\frac{K_{p}}{2}\left(n_0^2- \overline{n}(t)^2\right) \label{eq:diff-n},$$ where the constant $n_0>0$ is defined by $$\frac{1}{2}I_{\textrm{eff}} n_{0}^2 \equiv  E_{\textrm{tot}}.$$ This can be easily solved as $$\overline{n}(t) = n_{0}\frac{(n_{0} + n_{i})\exp(-n_{0}K_{p}t) - (n_{0} -n_{i}) }{(n_{0} + n_{i})\exp(-n_{0}K_{p}t) + (n_{0} -n_{i})}, 
\label{eq:gh-solution-p}$$ and we obtain the time for reversal $t_{rGH+}$ for the $n_{i} > 0$ case as $$t_{rGH+} 