Some extensions of the Einstein-Dirac equation

We considered an extension of the standard functional for the Einstein-Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler-Lagrange equations provide a new type of Einstein-Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein-Dirac system called the {\it CL-Einstein-Dirac equation of type II} (see Definition 3.1).