Commutative subalgebras from Serre relations.

We demonstrate that commutativity of numerous one-dimensional subalgebras in W 1 + ∞ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in algebra known as Serre relations. No other relations are needed for commutativity. The Serre relations survive the deformation to the affine Yangian Y ( gl ˆ 1) , hence the commutative subalgebras do as well. A special case of the Yangian parameters corresponds to the β -deformation. The preservation of Serre relations can be thought of a selection rule for proper systems of commuting β -deformed Hamiltonians. On the contrary, commutativity in the extended family associated with "rational (non-integer) rays" is not reduced to the Serre relations, and uses also other relations in the W 1 + ∞ algebra. Thus their β -deformation is less straightforward. [ABSTRACT FROM AUTHOR]