Killing vector fields for a special class of metrics

A one parameter local group of isometries of Riemannian or more generally pseudo-Riemannian manifolds is generated by a Killing vector field which is subjected to the commonly named Killing equations. The latter constitute an over determined system of first order partial differential equations which are even linear and homogeneous. However, in general such a system is not completely integrable.; A brief presentation of the well-known results on the existence of non-trivial Killing vector fields (i.e. nontrivial solutions of Killing equations) is provided. These results also suggest a method of constructing Killing vector fields, which consists basically of studying consequences of the so called integrability conditions. That last part requires usually quite involved symbolic computations and therefore can be aided by the appropriate computer programs.; The method is applied to a class of pseudo Riemannian structures that depends on two arbitrary holomorphic functions of one complex variable. Some constraints on these functions arise as a consequence of the existence of nontrivial Killing vector fields. The nature of these constraints and an explicit form of a Killing field are presented as the final result.