What $\xi$? Cosmological constraints on the non-minimal coupling
  constant
  In dynamical system describing evolution of universe with the flat
Friedmann-Robertson-Walker symmetry filled with barotropic dust matter and
non-minimally coupled scalar field with a constant potential function an
invariant manifold of the de Sitter state is used to obtain exact solutions of
the reduced dynamics. Using observational data coming from distant supernovae
type Ia, the Hubble function $H(z)$ measurements and information coming from
the Alcock-Paczy$\'n$ski test we find cosmological constraints on the
non-minimal coupling constant $\xi$ between the scalar curvature and the scalar
field. For all investigated models we can exclude negative values of this
parameter at the $68\%$ confidence level. We obtain constraints on the
non-minimal coupling constant consistent with condition for conformal coupling
of the scalar field in higher dimensional theories of gravity.
