There are various models of gravitation: the metrical Hilbert-Einstein theory, a wide class of intrinsically Lorentz-invariant tetrad theories (generally covariant in the space-time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, , etc). The gauge models are usually preferred but nevertheless it is an interesting idea to develop the class of -invariant (or rather -invariant) tetrad (n-leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus concept of affine symmetry. Formally, the obtained scheme is a generally covariant tetrad (n-leg) model, but it turns out that generally covariant and intrinsically affinely invariant models must have a kind of nonaccidental Born-Infeld-like structure. Let us also mention that they, being based on tetrads (n-legs), have many features common with continuous defect theories. It is interesting that they possess some group-theoretical solutions and more general spherically symmetric solutions, discussion of which is the main new result presented in this paper, including the applications of the 't Hooft-Polyakov monopoles in the generally covariant theories, which enables us to find some rigorous solutions of our strongly nonlinear equations. It is also interesting that within such a framework, the normal-hyperbolic signature of the space-time metric is not introduced by hand but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born-Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones.