If an analytic geometric structure is rigid in the sense of Gromov (see Gromov and d'Ambra), its symmetry vector fields form a finite dimensional Lie algebra. This is not very helpful, since we don't know too many different ways to prove rigidity, but there are lots of examples in:

D'Ambra, G.(F-IHES); Gromov, M.(I-CAGL)
*Lectures on transformation groups: geometry and dynamics.* **Surveys in differential geometry (Cambridge, MA, 1990)**, 19–111, Lehigh Univ., Bethlehem, PA, 1991. 

For example, if a structure (for example, a pseudo-Riemannian metric) induces an affine connection (for example, the Levi--Civita connection), then it is rigid, and its Lie algebra of symmetry vector fields is finite dimensional.
Similarly for a projective connection, or (in dimension 3 or more) for a conformal connection. 

You could also look at:

M. Gromov, *Rigid transformations groups*, in **Géométrie Différentielle** (Paris, 1986), Hermann, 1988.

A. M. Amores, *Vector fields of a finite type G-structure*, **J. Diff.
Geom.**, 1979.

R. Quiroga-Barranco; A. Candel, *Rigid and finite type geometric structures*. **Geom. Dedicata** 106 (2004), 123–143. 

R. Zimmer, *Ergodic theory and the automorphism group of a $G$-structure*, in **Group representations, ergodic theory, operator algebras,
and mathematical physics** (Berkeley, Calif., 1984), 1987.

The paper of Quiroga-Barranco and Candel explains how to prove rigidity of many types of geometric structures, including Cartan geometries modelled on effective homogeneous spaces, so including pseudo-Riemannian geometries, affine connections, conformal connections in dimension 3 or more, and projective connections in dimensions 2 or more.