= Solution
Rotational invariance gives $\{m_y,H\}=\{m_z,H\}=0$. Direct calculation gives
$$
m_x=\{m_y,m_z\}=yp_z-zp_y.
$$
Take $G=m_x^2+m_y^2+m_z^2$; angular-momentum brackets imply $\{G,m_x\}=0$, and central symmetry gives $\{G,H\}=0$. Hence $(H,G,m_x)$ are in involution.
At $\mathbf x=(1,0,0)$, $m_y=-p_z$, $m_z=p_y$. The differentials of $G$ and $m_x$ have independent components in the $p_y,p_z$ and transverse position directions when $p_y,p_z\ne0$; $dH$ supplies an independent radial/<momentum> component. Thus they are locally independent and the system is Liouville integrable.
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