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If i correctly understand your question, i think what you are talking about is the so called Poincare reduction method. This actually generalises Liouville integrability, in the sense that in the presence of involutive integrals of motion, it reduces the degrees of freedom but -in general- does not fully integrate the system.
Poincare's reduction (and thus Liouville's integrability) have been generalized for non-commutative, even non-Lie algebras of the integrals of motion via the Lie-Cartan theorem.
You can find details and examples at
Mathematical Aspects of Classical and Celestial Mechanics, vol. III, sect. 3.2.2, p.116-120.
See in particular, proposition 3.2, theorem 3.16 and examples 3.12, 3.13, for demonstrations of the methods for point particles in central fields.

Edit: the reduction method implied by the Lie-Cartan theorem, is more general, than the Poincare reduction method, in the sense that is applies not only for systems possesing integrals of motion in involution, (i.e. functions $F_i$ of the phase space coordinates which form abelian Lie algebras $\{F_i,F_j\}=0$ under the Poisson bracket) but also to dynamical systems possessing integrals whose algebras can be described in terms of generators and relations of the form $\{F_i,F_j\}=a_{ij}(F_k)$, where $a_{ij}$ are generally non-linear functions. (some times these can be shown to be infinite dimensional lie algebras).
I think there is a newer and even more general result on non-commutative integrability, by Mischenko and Fomenko but i do not have the exact reference right now. (i'll try to find it).

If i correctly understand your question, i think what you are talking about is the so called Poincare reduction method. This actually generalises Liouville integrability, in the sense that in the presence of $k$ involutive integrals of motion, it reduces the system to a $2n-k$ dim submanifold of the $2n$-dim phase space manifold; however -in general- does not fully integrate the system. (unless $k=n$, where we get Liouville integrability).

Poincare's reduction (and thus Liouville's integrability) have been generalized for non-commutative, even non-Lie algebras of the integrals of motion via the Lie-Cartan theorem.
You can find details and examples at
Mathematical Aspects of Classical and Celestial Mechanics, vol. III, sect. 3.2.2, p.116-120.
See in particular, proposition 3.2, theorem 3.16 and examples 3.12, 3.13, for demonstrations of the methods for point particles in central fields.

Edit: the reduction method implied by the Lie-Cartan theorem, is more general, than the Poincare reduction method, in the sense that it applies not only for systems possesing integrals of motion in involution, (i.e. functions $F_i$ of the phase space coordinates which form abelian Lie algebras $\{F_i,F_j\}=0$ under the Poisson bracket) but also to dynamical systems possessing integrals whose algebras can be described in terms of generators and relations of the form $\{F_i,F_j\}=a_{ij}(F_k)$, where $a_{ij}$ are generally non-linear functions. (some times these can be shown to be infinite dimensional lie algebras).
I think there is a newer and even more general result on non-commutative integrability, by Mischenko and Fomenko but i do not have the exact reference right now. (i'll try to find it).

If i correctly understand your question, i think what you are talking about is the so called Poincare reduction method. This actually generalises Liouville integrability, in the sense that it reduces the degrees of freedom but -in general- does not fully integrate the system.
Poincare's reduction (and thus Liouville's integrability) have been generalized for non-commutative, even non-Lie algebras of the integrals of motion via the Lie-Cartan theorem.
You can find details and examples at
Mathematical Aspects of Classical and Celestial Mechanics, vol. III, sect. 3.2.2, p.116-120.
See in particular, proposition 3.2, theorem 3.16 and examples 3.12, 3.13, for demonstrations of the methods for point particles in central fields.

Edit: the reduction method implied by the Lie-Cartan theorem, is more general, than the Poincare reduction method, in the sense that is applies not only for systems possesing integrals of motion in involution, (i.e. functions $F_i$ of the phase space coordinates which form abelian Lie algebras $\{F_i,F_j\}=0$ under the Poisson bracket) but also to dynamical systems possessing integrals whose algebras can be described in terms of generators and relations of the form $\{F_i,F_j\}=a_{ij}(F_k)$, where $a_{ij}$ are generally non-linear functions. (some times these can be shown to be infinite dimensional lie algebras).
I think there is a newer and even more general result on non-commutative integrability, by Mischenko and Fomenko but i do not have the exact reference right now. (i'll try to find it).

If i correctly understand your question, i think what you are talking about is the so called Poincare reduction method. This actually generalises Liouville integrability, in the sense that in the presence of involutive integrals of motion, it reduces the degrees of freedom but -in general- does not fully integrate the system.
Poincare's reduction (and thus Liouville's integrability) have been generalized for non-commutative, even non-Lie algebras of the integrals of motion via the Lie-Cartan theorem.
You can find details and examples at
Mathematical Aspects of Classical and Celestial Mechanics, vol. III, sect. 3.2.2, p.116-120.
See in particular, proposition 3.2, theorem 3.16 and examples 3.12, 3.13, for demonstrations of the methods for point particles in central fields.

Edit: the reduction method implied by the Lie-Cartan theorem, is more general, than the Poincare reduction method, in the sense that is applies not only for systems possesing integrals of motion in involution, (i.e. functions $F_i$ of the phase space coordinates which form abelian Lie algebras $\{F_i,F_j\}=0$ under the Poisson bracket) but also to dynamical systems possessing integrals whose algebras can be described in terms of generators and relations of the form $\{F_i,F_j\}=a_{ij}(F_k)$, where $a_{ij}$ are generally non-linear functions. (some times these can be shown to be infinite dimensional lie algebras).
I think there is a newer and even more general result on non-commutative integrability, by Mischenko and Fomenko but i do not have the exact reference right now. (i'll try to find it).

If i correctly understand your question, i think what you are talking about is the so called Poincare reduction method. This actually generalises Liouville integrability, in the sense that it reduces the degrees of freedom but -in general- does not fully integrate the system.
Poincare's reduction (and thus Liouville's integrability) have been generalized for non-commutative, non-Lie algebras of the integrals of motion via the Lie-Cartan theorem.
You can find details and examples at
Mathematical Aspects of Classical and Celestial Mechanics, vol. III, sect. 3.2.2, p.116-120.
See in particular, proposition 3.2, theorem 3.16 and examples 3.12, 3.13, for demonstrations of the methods for point particles in central fields.

Edit: the reduction method implied by the Lie-Cartan theorem, is more general, than the Poincare reduction method, in the sense that is applies not only for systems possesing integrals of motion in involution, (i.e. functions $F_i$ of the phase space coordinates which form abelian Lie algebras $\{F_i,F_j\}=0$ under the Poisson bracket) but also to dynamical systems possessing integrals whose algebras can be described in terms of generators and relations of the form $\{F_i,F_j\}=a_{ij}(F_k)$, where $a_{ij}$ are generally non-linear functions. (some times these can be shown to be infinite dimensional lie algebras).
I think there is a newer and even more general result on non-commutative integrability, by Mischenko and Fomenko but i do not have the exact reference right now. (i'll try to find it).

If i correctly understand your question, i think what you are talking about is the so called Poincare reduction method. This actually generalises Liouville integrability, in the sense that it reduces the degrees of freedom but -in general- does not fully integrate the system.
Poincare's reduction (and thus Liouville's integrability) have been generalized for non-commutative, even non-Lie algebras of the integrals of motion via the Lie-Cartan theorem.
You can find details and examples at
Mathematical Aspects of Classical and Celestial Mechanics, vol. III, sect. 3.2.2, p.116-120.
See in particular, proposition 3.2, theorem 3.16 and examples 3.12, 3.13, for demonstrations of the methods for point particles in central fields.

Edit: the reduction method implied by the Lie-Cartan theorem, is more general, than the Poincare reduction method, in the sense that is applies not only for systems possesing integrals of motion in involution, (i.e. functions $F_i$ of the phase space coordinates which form abelian Lie algebras $\{F_i,F_j\}=0$ under the Poisson bracket) but also to dynamical systems possessing integrals whose algebras can be described in terms of generators and relations of the form $\{F_i,F_j\}=a_{ij}(F_k)$, where $a_{ij}$ are generally non-linear functions. (some times these can be shown to be infinite dimensional lie algebras).
I think there is a newer and even more general result on non-commutative integrability, by Mischenko and Fomenko but i do not have the exact reference right now. (i'll try to find it).