  Inspired by the definition of the non-Gaussian two-parametric continuous
variable analogue of an isotropic state introduced by Mi\v{s}ta et al. [Phys.
Rev. A, 65, 062315 (2002); arXiv:quant-ph/0112062], we propose to take the
Gaussian part of this state as an independent state by itself, which yields a
simple, but with respect to the correlation structure interesting example of a
two-mode Gaussian analogue of an isotropic state. Unlike conventional isotropic
states which are defined as a convex combination of a thermal and an entangled
density operator, the Gaussian version studied here is defined by a convex
combination of the corresponding covariance matrices and can be understood as
entangled pure state with additional Gaussian noise controlled by a mixing
probability. Using various entanglement criteria and measures, we study the
non-classical correlations contained in this state. Unlike the previously
studied non-Gaussian two-parametric isotropic state, the Gaussian state
considered here features a finite threshold in the parameter space where
entanglement sets in. In particular, it turns out that it exhibits an analogous
phenomenology as the finite-dimensional two-qubit isotropic state.
