  We construct a secular theory of a coplanar system of N-planets not involved
in strong mean motion resonances, and which are far from collision zones.
Besides the point-to-point Newtonian mutual interactions, we consider the
general relativity corrections to the gravitational potential of the star and
the innermost planet, and also a modification of this potential by the
quadrupole moment and tidal distortion of the star. We focus on hierarchical
planetary systems. A survey regarding model parameters (the masses, semi-major
axes, spin rate of the star) reveals a rich and non-trivial dynamics of the
secular system. Our study is focused on its equilibria. Such solutions
predicted by the classic secular theory, which correspond to aligned (mode I)
or anti-aligned (mode II) apsides, may be strongly affected by the
gravitational corrections. The so called true secular resonance, which is a new
feature of the classic two-planet problem discovered by Michtchenko & Malhotra
(2004), may appear in other, different regions of the phase space of the
generalized model. We found bifurcations of mode II which emerge new, yet
unknown in the literature, secularly unstable equilibria and a complex
structure of the phase space. These equilibria may imply secularly unstable
orbital configurations even for nitially moderate eccentricities. The point
mass gravity corrections can affect the long term-stability in the secular time
scale, which may directly depend on the age of the host star through its spin
rate. We also analyze the secular dynamics of the upsilon Andromede system in
the realm of the generalized model. Also in this case of the three-planet
system, new secular equilibria may appear.
