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.