properties hold:

(i) there exists a unique $\epsilon \in [\epsilon^-,\epsilon^+] = [
      ( c_{x_0}) / K^+,$ $( c_{x_0}) / K^- ]$, such that the solution $x(t)$ of  with initial condition $x(t_0) = x_0$ is $T$-periodic, and setting $\Phi=x \circ \pi^{-1}$, the relation $\varphi = \Phi(s)$ is a VHC for .

(ii) If $\tilde\mu:\Re \to (0,\infty)$ is chosen so that $K^+ / K^- <
      (\cos x_0)^{-1}$, then the VHC $\varphi=\Phi(s)$ in part (i) satisfies $\image(\Phi) \subset (\Phi^-,\Phi^+) \subset (0,\pi/2)$, where $\Phi^-=\cos^{-1}\big(\frac{K^+}{K^-} c_{x_0}\big)$ and $\Phi^+=\cos^{-1}\big(\frac{K^-}{K^+} c_{x_0}\big)$.

\[rem:mu\] [By choosing $\tilde \mu(t) \equiv \tilde a(t)$, we have $K^+ = K^- =1$, $\epsilon^+ = \epsilon^- = c_{x_0}$. Proposition \[prop:generator:periodic\_solutions\] implies that, for all $x_0 \in (0,\pi/2)$, setting $\tilde \delta(t) = \cos (x_0)
  \tilde a(t)$, the VHC generator produces a $T$-periodic solution $x(t)$ with image contained in $(0,\pi/2)$. As a matter of fact, the solution in question is $x(t) \equiv x_0$, so that the relation $\varphi =x_0$ is a VHC for . The bicycle subject to this VHC has a constant roll angle as it travels around $\C$. The proposition provides great flexibility in finding VHCs with the property that the roll angle is confined within the interval $(0,\pi/2)$. All such constraints are compatible with 