Every central-force field is integrable, right? In 3d, there are four independent first integrals, namely, the three components of the angular momentum, and the total energy. 
So by the Liouville theorem, it is integrable, right? 
 A: 
by the Liouville theorem, it is integrable, right? 

If you are asking whether every central force corresponds to an integrable system then the answer depends on whether you require the original version of Newton's first law or whether you mistakeningly think it is redundant and replace it with a different law.
If you don't require that objects at rest subject to no force stay at rest (as opposed to just instantaneously have zero acceleration as the second law requires) then you can have central forces where there is no Hamiltonian  flow to even be integrable. So questions about its integrability make no sense.
As an example let $r=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$ where $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are the positions of the two particles. You can let $V(r)=-e^C(r-b)^a$ and then when you take the gradient you get a central force. And we get the third law. You can impose the second law as well. But unless you impose the first law as well, the motion can violate the first law if they start out a distance $b$ away, both initially at rest, and $0<a<1.$
Thus there isn't a unique solution to the equations of motion, hence no Hamiltonian flow, hence it doesn't make sense to say it is integrable or not integrable.
So it seems like just being a central force is in no way related to making there be vector fields with nice properties since it can't make there be vector fields at all.
Let's assume you pick a system that does have unique solutions, and so does have a flow. To use Liouville you need your integrals to be independent, and the three components of angular momentum are not independent at the origin.
My advice, if you want to use Liouville only consider systems where you have a flow. Remove bad places (where your integrals aren't independent or your flow isn't defined) such as the origin and remove anything that can flow to it or from it in finite time. Then try to use Liouville if you can actually meet the requirements (so time independent and have enough constants be independent). Then use other techniques for the parts of phase space you removed and for other problems where your flow just wasn't specified.
That is the historical approach and it's what other people do. For instance people usually do 1d first and then when they do 3d they simply don't care about 3d problems that stay in a line (because that is 1d and they already did 1d). So they ignore the region of phase space with zero angular momentum. You can too. Knowing what to ignore to get something that Liouville works for might be too hard in the sense that if tiu can do that then maybe you don't need Liouville.
