Department of Theoretical Physics,

Odessa National University,

2 Dvorjanskaja Str., Odessa, 65100, UKRAINE

E-mail: ovp@dtp.odessa.ua

**Maxwell-Dirac-Yang-Mills-Einstein equations: the integrability
conditions for the particle-like models with spherically symmetric gravitational
fields**

**Abstract:**

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.