Maxwell-Dirac-Yang-Mills-Einstein equations: the integrability conditions for the particle-like models with spherically symmetric gravitational fields
In spherical coordinates the energy-momentum tensor for a massive or massless fermion field in one of its eigenstates or in a superposition of eigenstates comprises a spherically symmetric part and angular-dependent parts. We study several field models, in which these angular-dependent parts are compensated by analogous counterparts for different fields such as electromagnetic, Yang-Mills, fermion fields, or their combinations. As a result, the total energy-momentum tensor in the model is spherically symmetric. So the gravitational field in this kinds of models is spherically symmetric too, and the initial system of the corresponting partial differential equations for it reduces to one of ordinary differential equations for the radial variable. The integrability conditions for the last equations are investigated.