Integrable homogeneous potentials of degree $-1$ in the plane with small eigenvalues

Abstract

We give a complete classification of meromorphically integrable homogeneous potentials $V$ of degree $-1$ which are real analytic on ${\mathbb{R}}^2\setminus \{0\}$. In the more general case when $V$ is only meromorphic on an open set of an algebraic variety, we give a classification of all integrable potentials having a Darboux point $c$ with $V^{\prime }(c)=-c,c_1^2+c_2^2\ne 0$ and $Sp({\nabla }^{2}V(c))\subset \{-1,0,2\}$. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case $V^{\prime }(c)=0$.

40 pages, 42 references

References in corpus (6)

• A Fast Approach to Creative Telescoping
• Swinging Atwood's Machine: Experimental and Theoretical Studies
• A note on algebraic potentials and Morales-Ramis theory
• On some exceptional cases in the integrability of the three-body problem
• Finiteness of integrable $n$-dimensional homogeneous polynomial potentials
• Necessary conditions for partial and super-integrability of Hamiltonian systems with homogeneous potentia

Cited by in corpus (1)

• Meromorphically integrable homogeneous potentials with multiple Darboux points