For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be EinsteinWeyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of these dispersionless PDEs can be seen from the geometry of their formal linearizations.MSC: 35L70, 35Q75, 53C25, 53C80, 53Z05.

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant's version of the Bott-Borel-Weil theorem to show that this bound is in fact sharp in almost all complex and split-real cases by exhibiting (abstract) models. We explicitly compute all submaximal symmetry dimensions when G is any complex or split-real simple Lie group.Comment: 44 pages. In order to better emphasize our main results, we reorganized our manuscript and reduced excessive background material and some running examples. (Readers wishing more background can view the relevant sections in version 3.

We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as "master dispersionless systems" in four and three dimensions, respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note, we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar partial differential equations for a general anti-self-dual conformal structure in neutral signature. C 2015 AIP Publishing LLC. [http://dx

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Three-dimensional Veronese webs are one-parametric foliations of a 3-dimensional space M by surfaces such that their tangents at any point x form a Veronese curve in Gr 2 (T x M) = P(T * x M). They appeared in the study of bi-Hamiltonian systems in [10], see also [23] and the references therein. In [6] a correspondence between Veronese webs and three-dimensional Lorenzian Einstein-Weyl structures of hyper-CR type was established. The latter due to [4] are parametrized by the solutions of the hyper-CR equationUsing the one-to-one correspondence with Veronese webs, the hyper-CR Einstein-Weyl structures were shown by Dunajski and Kryński [6] to be also parametrized by the solutions of the dispersionless Hirota equationwhich was introduced and studied by Zakharevich [25]. Both equations above are integrable and the parameters a, b, c are constants, but we will show that they can be taken functions without destroying the integrability. This can be seen as an integrable deformation, similar to [16], though the symmetry is essentially reduced in this process. The symmetry pseudogroup becomes an equivalence group for the deformation family, which eliminates the functional parameters though leaves new integrable dispersionless equations.Here and below by "integrable" we mean those equations that possess a dispersionless Lax pair, and we also show, motivated by [7], that they possess Einstein-Weyl structures on solutions, thus representing these equations as reductions of the Einstein-Weyl equation, integrable by the twistor methods [11]. The introduced equations are not contact equivalent, but they all parametrize Veronese webs and, using this fact, we will construct Bäcklund transformations between these equations.The equations that arise are of four types: A, B, C, D. Those that are translationally invariant (the standard requirement for hydrodynamic integrability test) together with equation 1 and the universal hierarchy equation [18] are equivalent to the five equations of Ferapontov-Moss [8] introduced in the context of quadratic line complexes. Our equations however arise from partially integrable Nijenhuis operators on the way to describe Veronese webs as a variation of a construction of Zakharevich [25].We establish a correspondence between partially integrable Nijenhuis operators to the operator fields with vanishing Nijenhuis tensor, and deduce from this a realization of Veronese webs through solutions of equations of any type A, B, C, D. We compute several examples of realizations, which also provide some exact solutions to the corresponding dispersionless PDEs.We perceive that the above correspondence can be used as a link between bi-Hamiltonian finitedimensional integrable systems and dispersionless integrable PDE related to the Veronese webs. In 1

Abstract. For the Spencer δ-cohomologies of a symbolic system we construct a spectral sequence associated with a subspace. We calculate the sequence for the systems of Cohen-Macaulay type and obtain a reduction theorem, which facilitates computation of δ-cohomologies by reducing dimension of the system. Using this algebraic result we prove an efficient compatibility criterion for a system of two scalar non-linear PDEs on a manifold of any dimension in terms of (generalized) Mayer brackets.

By developing the Tanaka theory for rank 2 distributions, we completely classify classical Monge equations having maximal finite-dimensional symmetry algebras with fixed (albeit arbitrary) pair of its orders. Investigation of the corresponding Tanaka algebras leads to a new Lie-Bäcklund theorem. We prove that all flat Monge equations are successive integrable extensions of the Hilbert-Cartan equation. Many new examples are provided.