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:

MR1144526 (93d:58117) Reviewed
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.