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## Banach Center Publications

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## Geometric features of Vessiot–Guldberg Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$

### Volume 113 / 2017

Banach Center Publications 113 (2017), 243-262 MSC: Primary 17B66; Secondary 34A26, 53B30, 53B50. DOI: 10.4064/bc113-0-13

#### Abstract

This paper locally classifies finite-dimensional Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$ relative to an arbitrary pseudo-Riemannian metric. Several results about their geometric properties are detailed, e.g. their invariant distributions and induced symplectic structures. Findings are illustrated with two examples of physical nature: the Milne–Pinney equation and the projective Schrödinger equation on the Riemann sphere.

#### Authors

• M. M. LewandowskiDepartment of Mathematical Methods in Physics
University of Warsaw
ul. Pasteura 5
02-093 Warszawa, Poland
e-mail
• J. de LucasDepartment of Mathematical Methods in Physics
University of Warsaw
ul. Pasteura 5
02-093 Warszawa, Poland
e-mail

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