The study classifies and normalizes 3D gl-regular Nijenhuis operators.
problem Classifying and normalizing 3D gl-regular Nijenhuis operators.
method Classification and normal form proof.
result Proved A. Bolsinov's conjecture.
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This paper studies gl-regular Nijenhuis operators and their properties.
problem Characterizing and understanding gl-regular Nijenhuis operators.
method Analyzing the properties of gl-regular Nijenhuis operators and proving their existence in a coordinate system.
result Discoveries of normal forms for singular points and topological restrictions for gl-regular Nijenhuis operators on closed surfaces.
Researchers describe local properties of Haantjes operators.
problem Understanding Haantjes operators with vanishing torsion.
method Complete local description of gl-regular Haantjes operators.
result Complete local description of gl-regular Haantjes operators.
The paper studies symmetries and conservation laws of non-diagonalisable hydrodynamic systems.
problem Integrating non-diagonalisable hydrodynamic systems of partial differential equations.
method Analysis of gl-regular Nijenhuis operators, splitting Theorem for symmetries and conservation laws, relationship between symmetries and conservation laws.
result The system of partial differential equations is integrable in quadratures.
New integrable systems constructed for non-diagonal Killing tensors.
problem Constructing integrable Hamiltonian systems with quadratic momenta.
method Using Nijenhuis geometry and gl-regular Nijenhuis operators.
result Reproduces classical Stäckel construction and finds new systems for n≥3.
The study describes metrics geodesically compatible with Nijenhuis operators and their applications to integrable systems.
problem Geodesically compatible metrics and their applications to integrable systems.
method Describes metrics geodesically compatible with a gl-regular Nijenhuis operator and shows how these metrics relate to integrable PDE systems.
result Every metric geodesically compatible with a Nijenhuis operator gives a finite-dimensional reduction of an integrable PDE system.
The paper studies Nijenhuis operators with a unity and their connection to F-manifolds.
problem Understanding Nijenhuis operators and their relationship to F-manifolds.
method Established a Splitting Theorem for Nijenhuis operators with a unity and proved their equivalence to F-manifolds.
result The class of regular F-manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity.