The detection of the first exoplanet around a solar-type
star revealed the existence of close-in planets. Several of these
close-in planets are part of multi-planet systems. For systems
detected via the radial velocity (RV) method, we lack information on
the mutual inclination of the orbital planes. The aim of this work
is to study the long-term stability of RV-detected two-planet
systems with close-in planets and identify possible
three-dimensional configurations for these systems that are
compatible with observations. To do so, we focused on the protective
mechanism of the Lidov-Kozai (LK) secular resonance and studied the
effects of general relativity (GR) on long-term evolution. By
means of an analytical study based on a high-order secular
Hamiltonian expansion in the eccentricities and
inclinations, we first identified ranges of values for the orbital
and mutual inclinations that are compatible with the presence of the
LK resonance in the purely gravitational case. Then, adding the
secular contribution of the relativistic corrections exerted by the
central star on the inner planet, namely the advance of its
pericenter precession, we analysed the outcomes of the two sets of
simulations. We compared our results to analytical estimates to
determine the importance of GR effects. We find that for the
majority of the systems considered, GR strongly affects the dynamics
of the system and, most of the time, voids the LK resonance, as
observed for GJ $649$, GJ $832$, HD $187123$, HD $190360$,
HD $217107$, and HD $47186$. The long-term stability of these
systems is then possible whatever the mutual inclination of the
orbits. On the contrary, for GJ $682$, HD $11964$, HD $147018$, and
HD $9446$, the LK resonant region in the parameter space of the
orbital and mutual inclinations is left (almost) unchanged when GR
effects are considered, and consequently their long-term stability
is only possible if the mutual inclination of the orbits is low or
if the systems are in the LK regime with a high mutual
inclination.