Towards generic cosmological evolution without singularity and fundamental symmetry of space-time

Dynamical systems methods are used to investigate cosmological model with non-minimally coupled scalar field. Existence of an asymptotically unstable de Sitter state distinguishes values of the non-minimal coupling constant parameter $\frac{3}{16}\le \xi <\frac{1}{4}$, which correspond to conformal coupling in higher dimensional theories of gravity.