[For part I see the review above (Zbl 0556.70014).] The author finds a necessary condition for the existence of a sufficient number of algebraic first integrals for a class of dynamical systems with similarity invariance. In fact the author proves that in order that a given similarity invariant system with rational right-hand sides is algebraically integrable every possible Kowalevski's exponent must be some rational number. Then he applies the theorem to the classical \(n\)-body problem and proves that the three-body problem is not algebraically integrable. Finally he makes a statement for non-similarity invariant systems and shows algebraic non-integrability of the Henon-Heiles system as an example.