= Solution
For constant $r=R>0$, the radial equation requires $\dot\theta(\dot\theta+2e\mu/m)=0$. Select the stipulated nonzero <angular velocity> and define the positive <cyclotron frequency> $\Omega_c=2e\mu/m$. Then
$$
\boxed{\theta(t)=\theta_0-\Omega_ct,\qquad
z(t)=Z\cos(\Omega_zt+\delta),\qquad
\Omega_z=\sqrt{2e\lambda/m}.}
$$
This is a circle in the transverse plane combined with an independent axial <oscillation>, as in <uniform magnetic field with axial harmonic confinement>. Viewed from positive $z$, positive charge rotates clockwise. Its three-dimensional path lies on the cylinder $r=R$ and reverses its vertical motion; \b[it is generally not a constant-pitch helix]. For $Z=0$ it is a horizontal circle. This orbit has $p_\theta=-e\mu R^2$.
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