  The formalism for Poisson-Hopf (PH) deformations of Lie-Hamilton systems is
refined in one of its crucial points concerning applications, namely the
obtention of effective and computationally feasible PH deformed superposition
rules for prolonged PH deformations of Lie-Hamilton systems. The two new
notions here proposed are a generalization of the standard superposition rules
and the concept of diagonal prolongations for Lie systems, which are
consistently recovered under the non-deformed limit. Using a technique from
superintegrability theory, we obtain a maximal number of functionally
independent constants of the motion for a generic prolonged PH deformation of a
Lie-Hamilton system, from which a simplified deformed superposition rule can be
derived. As an application, explicit deformed superposition rules for prolonged
PH deformations of Lie-Hamilton systems based on the oscillator Lie algebra
${h}_4$ are computed. Moreover, by making use that the main structural
properties of the book subalgebra ${b}_2$ of ${h}_4$ are preserved under the PH
deformation, we consider prolonged PH deformations based on ${b}_2$ as
restrictions of those for ${h}_4$-Lie-Hamilton systems, thus allowing the study
of prolonged PH deformations of the complex Bernoulli equations, for which both
the constants of the motion and the deformed superposition rules are explicitly
presented.
