Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation

Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Mart{\'ı}nez Alonso--Shabat equation

“…So, if we choose the parameter λ, we substitute θ = n∈Z θ n λ n , ω = n∈Z ω n λ n , then collect the terms at λ n for a fixed arbitrary n ∈ Z, and finally rename θ n−1 → θ, θ n → θ, ω n−1 → ω, ω n → ω. This gives two systems (23) θx = −(κ t…”

“…Remark 1. From Systems (23) and (24) we have R κ,µ = (κ t+µ z +1) R 0,0 + (κ y + µ x) 1 and Rκ,µ = (κ t + µ z + 1) R0,0 + (κ y + µ x) 1, where 1 is the identical map on the spaces of shadows of Eq. ( 13) and its cotangent extension.…”

“…The Lax representation (21) as well as the recursion operator (23), (24) survive in these reductions when κ = µ = 0 and thus provide Lax representations and recursion operators for the cotangent extensions of the above reduced equations. The questions of whether these cotangent extensions possess parametric families of Lax representations and recursion operators, and whether the recursion operators are hereditary, are more subtle.…”

Infinitely Many Commuting Nonlocal Symmetries for Modified Martinez Alonso-Shabat Equation

“…which describes three-dimensional Veronese webs and is a subject of intense research, see e.g. [15,16] and references therein. Thus, (1) can be seen as a natural 4D generalization of (3) and hence of (4).…”

“…Interestingly, the situation appears to be quite different in 3D, where infinitedimensional noncommutative algebras of nonlocal symmetries for a number of dispersionless integrable systems were found by direct computations, see e.g. [8,9,15,22].…”