Nontrivial 1-parameter families of zero-curvature representations obtained via symmetry actions

On the (Non)Removability of Spectral Parameters in Z 2 $\mathbb{Z}_{2}$ -Graded Zero-Curvature Representations and Its Applications

“…We refer to [62] for discussion of the set-up when a symmetry X ∈ Γ (T E ∞ ) of an equation E can be used to extend a given zero-curvature representation for E to a "nontrivial" family of zero-curvature representations. We expect that this result must have a straightforward generalisation to the Z 2 -graded case.…”

Nonlocal conservation laws of the constant astigmatism equation

“…Below we shall be interested in how conservation laws change under various local and nonlocal symmetries of the equation. It is typical for an integrable equation that its symmetries either preserve the zero curvature representation or send it to a gaugeequivalent one (see [6] for an infinitesimal criterion). If the zero curvature representation is preserved, then so are the hierarchies.…”

Symmetry and Lie–Frobenius reduction of differential equations

“…( 13) obviously defines smooth local functions Λ i = Λ i (x, u, u x ), if we look for the A which satisfy (13) for given Λ i , it is much less obvious that a solution exists, and even less that A will be a local function of (x, u). The first fact is guaranteed by the (horizontal) Maurer-Cartan equation (10), but the second is in general not true.…”

“…This point will be made clearer by an explicit example. For two independent variables (x, y) and one dependent variable u, consider Λ x = 0 u x 0 0 ; one can check that the most general form of Λ y satisfying the horizontal Maurer-Cartan equation (10) is…”