= Solution
Let $\mathbf b=\mathbf B/B$ for a nonzero constant <magnetic field>. Since $\gamma$ is constant, the <relativistic Lorentz force> reduces to
$$
\dot{\mathbf v}=\frac e{m\gamma_0}\mathbf v\times\mathbf B.
$$
The projection along $\mathbf b$ is constant. In the perpendicular plane define the signed <cyclotron frequency> $\Omega=eB/(m\gamma_0)$; then
$$
\dot{\mathbf v}_\perp=\Omega\mathbf v_\perp\times\mathbf b,\qquad
\ddot{\mathbf v}_\perp=-\Omega^2\mathbf v_\perp.
$$
The explicit rotating velocity is
$$
\boxed{\mathbf v_\perp(t)=\mathbf v_{\perp0}\cos(\Omega t)
+(\mathbf v_{\perp0}\times\mathbf b)\sin(\Omega t),\qquad
T=\frac{2\pi\gamma_0m}{|e|B}.}
$$
This is the <relativistic cyclotron orbit> frequency in laboratory time, reduced from the nonrelativistic frequency by $1/\gamma_0$. With nonzero parallel velocity, the trajectory is a helix. If $\mathbf v_{\perp0}=0$, that perpendicular vector stays zero and has no nontrivial fundamental period; the displayed $T$ is the period of a nonzero transverse orbit.
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