Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Darboux points and integrability of homogeneous Hamiltonian systems with
  three and more degrees of freedom   We consider natural complex Hamiltonian systems with $n$ degrees of freedom
given by a Hamiltonian function which is a sum of the standard kinetic energy
and a homogeneous polynomial potential $V$ of degree $k>2$. The well known
Morales-Ramis theorem gives the strongest known necessary conditions for the
Liouville integrability of such systems. It states that for each $k$ there
exists an explicitly known infinite set $\scM_k\subset\Q$ such that if the
system is integrable, then all eigenvalues of the Hessian matrix $V''(\vd)$
calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$, belong to
$\scM_k$. The aim of this paper is, among others, to sharpen this result. Under
certain genericity assumption concerning $V$ we prove the following fact. For
each $k$ and $n$ there exists a finite set $\scI_{n,k}\subset\scM_k$ such that
if the system is integrable, then all eigenvalues of the Hessian matrix
$V''(\vd)$ belong to $\scI_{n,k}$. We give an algorithm which allows to find
sets $\scI_{n,k}$. We applied this results for the case $n=k=3$ and we found
all integrable potentials satisfying the genericity assumption. Among them
several are new and they are integrable in a highly non-trivial way. We found
three potentials for which the additional first integrals are of degree 4 and 6
with respect to the momenta.

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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: math
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex