= Solution
A <Hamiltonian system> with $n$ <degrees of freedom> is <Liouville integrable> if it has $n$ functionally independent <first integrals>, including the <Hamiltonian>, whose pairwise <Poisson brackets> vanish. <Functional independence> is required on a regular open set. The <Arnold-Liouville theorem> says that a connected compact regular common level of these integrals is an $n$-torus, with a neighborhood admitting <action-angle coordinates>; the <Hamiltonian> flow there has constant angular velocities and fixed actions. Compactness is essential to the torus conclusion.
Use $\{f,g\}=\sum_j(f_{q_j}g_{p_j}-f_{p_j}g_{q_j})$. On $r>0$, for differentiable $F$, <Hamilton's equations> give $\dot q=p$, $\dot p=-F'(r)q/r$. Hence the <angular momentum> $M=q\times p$ obeys
$$
\dot M=\dot q\times p+q\times\dot p=p\times p-\frac{F'(r)}r q\times q=0.
$$
In particular $M_1$ and $M_2$ are <first integrals>. The <Jacobi identity for the Poisson bracket> is
$$
\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0.
$$
Taking $f=H,g=M_1,h=M_2$ shows $\{H,\{M_1,M_2\}\}=0$. Direct differentiation gives $M_3=\{M_1,M_2\}=q_1p_2-q_2p_1$ and, cyclically, $\{M_i,M_j\}=\epsilon_{ijk}M_k$.
Set $L^2=M_1^2+M_2^2+M_3^2$. Then $\{L^2,M_3\}=2M_1(-M_2)+2M_2M_1=0$, and the brackets of $H$ with both expressions vanish. Thus
$$
\boxed{H,\ L^2,\ M_3\text{ are three commuting first integrals}.}
$$
For <functional independence>, at $q=(r,0,0)$ with $r>0$ the <Jacobian matrix> of $(H,L^2,M_3)$ with respect to $(p_1,p_2,p_3)$ is
$$
\begin{pmatrix}p_1&p_2&p_3\\0&2r^2p_2&2r^2p_3\\0&r&0\end{pmatrix},
$$
whose <determinant> is $-2r^3p_1p_3$. It is nonzero when $p_1p_3\ne0$. More generally this minor is a nonzero polynomial in $q,p$, independent of $F$, so its zero set has empty interior and the integrals are independent on a dense open regular set. This proves <Liouville integrability> on the regular set. The arbitrary potential in the question does not ensure compact common levels: $F=0$ has unbounded free trajectories. The full torus conclusion applies to those regular connected common levels which are compact, rather than to every choice of $F$ and every level.
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