  In this paper we exploit the phenomenon of two principal half eigenvalues in
the context of fully nonlinear Lane-Emden type systems with possibly unbounded
coefficients and weights. We show that this gives rise to the existence of two
principal spectral curves on the plane. We also construct a possible third
spectral curve related to a second eigenvalue and an anti-maximum principle,
which are novelties even for Lane-Emden systems involving linear operators. As
applications, we derive a maximum principle in small domains for these systems,
as well as existence and uniqueness of positive solutions in the sublinear
regime. Most of our results are new even in the scalar case, in particular for
a class of Isaac's operators with unbounded coefficients, whose $W^{2,\varrho}$
regularity estimates we also prove.
