Dirac like equations and generalized Majorana fields, intrinsic symmetries

We start with the the multicomponent matrix equation $(\Gamma_\mu\partial_\mu+m)\psi=0$, and introduce the concept of the intrinsic symmetry. These symmetries should preserve the form of the basic equation. The relevant Lagrangian should be invariant under the intrinsic symmetry transformation. We will impose one additional requirement on symmetry transformations: such transformations should preserve the Majorana nature of the fields. This means that if the function $\Psi_A$ is real (imaginary) part of the wave function, then after symmetry transformation the function remains real (imaginary). The situation for massless field $\Gamma_\mu\partial_\mu\psi=0$ is substantially different. The Lagrangian invariance with respect to intrinsic symmetry transformation for massless case coincide with that for massive case. The main accent will be done on multicomponent Majorana fields, which can be related to one, two, three and four Dirac fields.