  The paper considers a slightly modified one-dimensional infinite mass-in-mass
chain. In the case of the long-wave approximation, which corresponds to the
transition to a continuous medium, we obtained a system of two equations, which
is a generalization of the classical mechanics Klein-Gordon-Fock equation and
has both optical and acoustic branches of the dispersion relation. Based on
this classical mechanics system of equations, we have proposed a system of two
relativistic quantum mechanics equations, which is a generalization of the
relativistic quantum mechanics Klein-Gordon-Fock equation. Next, based on this
system and following the Dirac approach, we have proposed the generalization of
the Dirac equation for a free electron with an eight-component wave function in
the form of a system of eight linear partial differential equations of the
first order. Unlike the Dirac equation with a four-component wave function,
which has only an optical branch of the dispersion relation, the generalized
Dirac equation has both optical and acoustic branches of the dispersion
relation, each of which has two branches with positive and negative energies,
respectively. We have calculated phase and group velocities for all cases. For
the positive and negative acoustic branches, the phase and group velocities are
equal in modulus to the speed of light. For the positive and negative optical
branches, the phase and group velocities have a structure like that of de
Broglie waves. In the one-dimensional case, eight linearly independent
solutions corresponding to eight combinations of two branches of dispersion,
two signs of total energy, and two possible directions of spin orientation,
each in the form of four plane waves, are obtained.
