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We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.
We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic
field which is described by the magnetic Schrödinger operator with a periodic potential
plus a finitely supported perturbation. We describe all eigenvalues and resonances of this
operator, and theirs dependence on the magnetic field. The proof is reduced to the
analysis of the periodic Jacobi operators on the half-line with finitely supported
perturbations.
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