Abstract
Some open problems appearing in the primary article on the symmetry reduction are solved. A new and quite simple coordinate-free definition of Poincare-Cartan forms and the substance of divergence symmetries (quasisymmetries) are clarified. The unbeliavable uniqueness and therefore the global existence of Poincare-Cartan forms without any uncertain multipliers for the Lagrange variational problems are worth extra mentioning.
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