Symmetry and Lie-Frobenius reduction of differential equations

Twisted symmetries, widely studied in the last decade, proved to be as effective as standard ones in the analysis and reduction of nonlinear equations. We explain this effectiveness in terms of a Lie-Frobenius reduction; this requires to focus not just on the prolonged (symmetry) vector fields but on the distributions spanned by these and on systems of vector fields in involution in Frobenius sense, not necessarily spanning a Lie algebra.