We construct the most general known classical, nonlinear and nonlocal generalization of the conventional Galilei transformations, as well as the corresponding classical realization of the infinite family of Galilei-isotopic symmetries G-circumflex (3.1) proposed in preceding works, under the condition that they result to be all locally isomorphic to the conventional Galilei symmetry. The symmetries G-circumflex (3.1) are then used to characterize the largest possible class of nonlinear, nonlocal and nonhamiltonian Newtonian systems which still verify the conservation laws of all ten, total, physical quantities, as preparatory grounds for subsequent operator studies for the hadronic structure. The method for the explicit construction of the space-time isosymmetries G-circumflex (3.1) from given Galilei-noninvariant equations of motion is outlined. (author). 11 refs.