Russian Gravitational Society Institute of Metrological Service Center for Gravitation and Fundamental Metrology

We consider multidimensional cosmologies in even-dimensional space-times (D = 2n) containg perfect fluid and a multidimensional generalization of the Maxwell field FA1···An preserving its conformal invariance. Among models with an isotropic physical 3-space some integrable cases are found: vacuum models (which are integrable in the general case) and some perfect fluid models with barotropic equations of state. All of them contain a component of the F field appearing as an additional scalar in 4 dimensions. A two-parameter family of spatially flat models and four one-parameter families, including non-spatially flat models, have been obtained (where the parameters are constants from the fluid equation of state). All these integrable models admit the inclusion of a massless scalar field or an additional fluid with the maximally stiff equation of state. Basic properties of vacuum models in the physical conformal frame are outlined.