  Qualitative characteristics and the rigorous definition of a concept of the
double symmetry is given. We use some double symmetry for constructing a theory
of fields not investigated before which transform as the proper Lorentz group
representations decomposable into infinite direct sums of finite-dimentional
irreducible representations. All variants of the double-symmetric free
Lagrangian of such fields are in brief described. The solution of the problem
of changing Lagrangian mass terms due to a spontaneous breaking of the
secondary symmetry is stated. The general properties of the mass spectra of
fermions in the considered theory are given. It is pointed out a region of free
parameters where the theoretical mass spectra qualitatively correspond to a
picture typical for the parton model of hadrons.
