We study the Lorentzian manifolds M 1 , M 2 , M 3 , and M 4 obtained by small changes of the standard Euclidean metric on ℝ 4 with the punctured origin O. The spaces M 1 and M 4 are closed isotropic space-time models. The manifolds M 3 and M 4 (respectively, M 1 and M 2 ) are geodesically (non)complete; M 1 are M 4 are globally hyperbolic, while M 2 and M 3 are not chronological. We found the Lie algebras of isometry and homothety groups for all manifolds; the curvature, Ricci, Einstein, Weyl, and energy-momentum tensors. It is proved that M 1 and M 4 are conformally flat, while M 2 and M 3 are not conformally flat and their Weyl tensor has the first Petrov type.