  We develop a theory of a quantifier of bipartite Gaussian entanglement called
Gaussian intrinsic entanglement (GIE) which was proposed recently in [L.
Mi\v{s}ta {\it et al.}, Phys. Rev. Lett. {\bf 117}, 240505 (2016)]. The GIE
provides a compromise between computable and physically meaningful entanglement
quantifiers and so far it was calculated for two-mode Gaussian states including
all symmetric partial minimum-uncertainty states, weakly mixed asymmetric
squeezed thermal states with partial minimum uncertainty, and weakly mixed
symmetric squeezed thermal states. We improve the method of derivation of GIE
and we show, that all previously derived formulas for GIE of weakly mixed
states in fact hold for states with higher mixedness. In addition, we derive
analytical formulas for GIE for several new classes of two-mode Gaussian states
with partial minimum uncertainty. Finally, it is shown, that like for all
previously known states, also for all new states the GIE is equal to Gaussian
R\'{e}nyi-2 entanglement of formation. This finding strengthens a conjecture
about equivalence of GIE and Gaussian R\'{e}nyi-2 entanglement of formation for
all bipartite Gaussian states.
