Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Residual symmetries and B\"acklund transformations   It is proved that for a given truncated Painlev\'e expansion of an arbitrary
nonlinear Painlev\'e integrable system, the residue with respect to the
singularity manifold is a nonlocal symmetry. The residual symmetries can be
localized to Lie point symmetries after introducing suitable prolonged systems.
The finite transformations of the residual symmetries are equivalent to the
second type of Darboux-B\"acklund transformations. The once B\"acklund
transformations related to the residual symmetries are same for many integrable
systems including the Korteweg-de Vries, Kadomtsev-Petviashvili, Boussinesq,
Sawada-Kortera and Kaup-Kupershmidt equations. For the Korteweg-de Vries
equation, the $n^{th}$ Darboux transformations can also be obtained from the
Lie point symmetry approach via the localization of the residual symmetries.

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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom