Several physical problems such as the "twin paradox" in curved spacetimes have a purely geometrical nature and are reduced to studying properties of bundles of timelike geodesics. The paper is a general introduction to systematic investigations of the geodesic structure of physically relevant spacetimes. These are focussed on the search of locally maximal timelike geodesics. The method is based on determining conjugate points on chosen geodesic curves. The method presented here is effective at least in the case of radial and circular geodesics in static spherically symmetric spacetimes. Our approach shows that even in Schwarzschild spacetime (as well as in other static spherically symmetric ones), one can find a new unexpected geometrical feature: each stable circular orbit contains besides the obvious set of conjugate points two other sequences of conjugate points. The obvious limitations of the approach arise from one's inability to solve involved ordinary differential equations and the recent progress in the field allows one to increase the range of metrics and types of geodesic curves tractable by this method.

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The Szekeres system is a four-dimensional system of first-order ordinary differential equations with nonlinear but polynomial (quadratic) right-hand side. It can be derived as a special case of the Einstein equations, related to inhomogeneous and nonsymmetrical evolving spacetime. The paper shows how to solve it and find its three global independent first integrals via Darboux polynomials and Jacobi's last multiplier method. Thus the Szekeres system is completely integrable. Its two-dimensional subsystem is also investigated: we present its solutions explicitly and discuss its behaviour at infinity.

In the context of the classical Kaluza–Klein cosmology the generalized Bianchi models in 11 dimensions are considered. These are space-times whose spacelike ten-dimensional sections are the hypersurfaces of transitivity for a ten-dimensional isometry group of the total space-time. Such a space-time is a trivial principal fiber bundle P(M,G7), where M is a four-dimensional physical space-time with an isometry group G3 (of a Bianchi type) and G7 is a compact isometry group of the compact internal space. The isometry group of P is G10=G3⊗G7, hence all the generalized Bianchi models are classified by enumerating the relevant groups G7. Due to the compactness of G7 the result is astonishingly simple: there are three distinct homogeneous internal spaces in addition to the 11 ordinary Bianchi types for M.