  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).
