After reviewing some notions of the formal theory of diierential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the eeects of local solvability and of gauge symmetries, respectively. We consider non-classical symmetry reductions and more general reductions using diierential constraints.

Theorem (1) There is an involution C of G satisfying: C(h) = h −1 (h ∈ H); (2) C(g) ∼ g −1 for all semisimple elements g; (3) Any two such involutions are conjugate by an inner automorphism;

It has been recently proved that a variety of associative PI-superalgebras with graded involution finite basic rank over field characteristic zero is minimal fixed ⁎-graded exponent if, and only it generated by subalgebra an upper block triangular matrix algebra, A:=UTZ2⁎(A1,…,Am), equipped suitable elementary Z2-grading involution. Here we give necessary sufficient conditions so IdZ2⁎(A) facto...

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point real Lagrangian submanifold in closed spherically monotone symplectic manifold must be even. As corollary, we exhibit genuine phenomenon terms involutions, namely Chekanov torus $\mathbb{T}_{\text{Chek}}$ $S^2\times S^2$, which is not Hamiltonian isotopic to Clifford $\mathbb{T}_{\text{Clif}}$, can s...

In the present paper we study generalized left derivations on Lie ideals of rings with involution. Some of our results extend other ones proven previously just for the action of generalized left derivations on the whole ring. Furthermore, we prove that every generalized Jordan left derivation on a 2-torsion free ∗-prime ring with involution is a generalized left derivation.