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The formalism for Poisson-Hopf (PH) deformations of Lie-Hamilton systems isrefined in one of its crucial points concerning applications, namely theobtention of effective and computationally feasible PH deformed superpositionrules for prolonged PH deformations of Lie-Hamilton systems. The two newnotions here proposed are a generalization of the standard superposition rulesand the concept of diagonal prolongations for Lie systems, which areconsistently recovered under the non-deformed limit. Using a technique fromsuperintegrability theory, we obtain a maximal number of functionallyindependent constants of the motion for a generic prolonged PH deformation of aLie-Hamilton system, from which a simplified deformed superposition rule can bederived. As an application, explicit deformed superposition rules for prolongedPH deformations of Lie-Hamilton systems based on the oscillator Lie algebra${h}_4$ are computed. Moreover, by making use that the main structuralproperties of the book subalgebra ${b}_2$ of ${h}_4$ are preserved under the PHdeformation, we consider prolonged PH deformations based on ${b}_2$ asrestrictions of those for ${h}_4$-Lie-Hamilton systems, thus allowing the studyof prolonged PH deformations of the complex Bernoulli equations, for which boththe constants of the motion and the deformed superposition rules are explicitlypresented.

Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra