  In this paper, we employ a combination of analytical and numerical techniques
to investigate the dynamics of lattice envelope vector soliton solutions
propagating within a one-dimensional chain of flexible mechanical metamaterial.
To model the system, we formulate discrete equations that describe the
longitudinal and rotational displacements of each individual rigid unit mass
using a lump element approach. By applying the multiple-scales method in the
context of a semi-discrete approximation, we derive an effective nonlinear
Schr\"odinger equation that characterizes the evolution of rotational and
slowly varying envelope waves from the aforementioned discrete motion
equations. We thus show that this flexible mechanical metamaterial chain
supports envelope vector solitons where the rotational component has the form
of either a bright or a dark soliton. In addition, due to nonlinear coupling,
the longitudinal displacement displays kink-like profiles thus forming the
2-components vector soliton. These findings, which include specific vector
envelope solutions, enrich our knowledge on the nonlinear wave solutions
supported by flexible mechanical metamaterials and open new possibilities for
the control of nonlinear waves and vibrations.
