Poles of pseudo-Riemannian spaces

Two-dimensional complete analytic pseudo-Riemannian spaces $V$ with poles are studied. A pole is a point $p\in V$ with respect to which $V$ admits a one-parameter group of rotations. With each pole is connected a holomorphic function $F_p(z)$ (the complex pole function). Necessary conditions on $F_p(z)$ are established. A number of “existence theorems” are proved: for a given holomorphic function $F(z)$ with certain properties there exists a complete space $V$ with pole $p$ for which the function $F_p(z)$ coincides with $F(z)$.