  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 Schr\"odinger equation on the Riemann
sphere.
