  We study which geometric structure can be constructed from the vierbein
(frame/coframe) variables and which field models can be related to this
geometry. The coframe field models, alternative to GR, are known as viable
models for gravity, since they have the Schwarzschild solution. Since the local
Lorentz invariance is violated, a physical interpretation of additional six
degrees of freedom is required. The geometry of such models is usually given by
two different connections -- the Levi-Civita symmetric and metric-compatible
connection and the Weitzenbock flat connection. We construct a general family
of linear connections of the same type, which includes two connections above as
special limiting cases. We show that for dynamical propagation of six
additional degrees of freedom it is necessary for the gauge field of
infinitesimal transformations (antisymmetric tensor) to satisfy the system of
two first order differential equations. This system is similar to the vacuum
Maxwell system and even coincides with it on a flat manifold. The corresponding
``Maxwell-compatible connections'' are derived. Alternatively, we derive the
same Maxwell-type system as a symmetry conditions of the viable models
Lagrangian. Consequently we derive a nontrivial decomposition of the coframe
field to the pure metric field plus a dynamical field of infinitesimal Lorentz
rotations. Exact spherical symmetric solution for our dynamical field is
derived. It is bounded near the Schwarzschild radius. Further off, the solution
is close to the Coulomb field.
