As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.

Two Poisson brackets $\{ \cdot,\cdot \} _{1} , \{ \cdot , \cdot \} _{2}$ on a manifold $M$ are compatible if their arbitrary linear combination $\lambda \{ \cdot , \cdot \} _1+\mu\{\cdot,\cdot\} _2$ is also a Poisson bracket. A bi-Hamiltonian system is one which allows Hamiltonian formulations with respect to two compatible Poisson brackets. It automatically posseses a number of integrals in involution.

The definition of a complete integrability (à la Liouville-Arnol'd) is:

Hamiltonian flows and Poisson maps on a $2n$-dimensional symplectic manifold $\left(M,\{ \cdot, \cdot \}_M\right)$ with $n$ (smooth real valued) functions $F _1,F _2,\dots,F _n$ such that: (i) they are functionally independent (i.e. the gradients $\nabla F _k$ are linearly independent everywhere on $M$) and (ii) these functions are in involution (i.e. $\{F _k,F _j\}=0$) are called completely integrable.

Now, I would like to understand the connections between these two notions, and because I haven't studied the theory, any answer would be helpful. I find reading papers on these subjects too technical at the moment. Specific questions I have in mind are:

Does completely integrable system always allow for a bi-Hamiltonian structure? Is every bi-Hamiltonian system completely integrable? If not, what are examples (or places where to find examples) of systems that posses one property but not the other?

I apologize for any stupid mistakes I might have made above. Feel free to edit (tagging included).

As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.

Two Poisson brackets $\{ \cdot,\cdot \} _{1} , \{ \cdot , \cdot \} _{2}$ on a manifold $M$ are compatible if their arbitrary linear combination $\lambda \{ \cdot , \cdot \} _1+\mu\{\cdot,\cdot\} _2$ is also a Poisson bracket. A bi-Hamiltonian system is one which allows Hamiltonian formulations with respect to two compatible Poisson brackets. It automatically posseses a number of integrals in involution.

The definition of a complete integrability (à la Liouville-Arnol'd) is:

Hamiltonian flows and Poisson maps on a $2n$-dimensional symplectic manifold $\left(M,\{ \cdot, \cdot \}_M\right)$ with $n$ (smooth real valued) functions $F _1,F _2,\dots,F _n$ such that: (i) they are functionally independent (i.e. the gradients $\nabla F _k$ are linearly independent everywhere on $M$) and (ii) these functions are in involution (i.e. $\{F _k,F _j\}=0$) are called completely integrable.

Now, I would like to understand the connections between these two notions, and because I haven't studied the theory, any answer would be helpful. I find reading papers on these subjects too technical at the moment. Specific questions I have in mind are:

Does completely integrable system always allow for a bi-Hamiltonian structure? Is every bi-Hamiltonian system completely integrable? If not, what are examples (or places where to find examples) of systems that posses one property but not the other?

I apologize for any stupid mistakes I might have made above. Feel free to edit (tagging included).