Invariants and solutions of Gurzadyan-Xue dark energy cosmological models

We have derived the invariants of cosmological models with Gurzadyan-Xue dark energy, along with the solutions for any time dependent light speed and gravitational constant. The correspondence of the invariants with the separatrices found earlier for the GX-models in \cite{Ver06a} is shown, and hence the basis of then detected hidden symmetry is now revealed. Solutions are derived both for radiation and matter models, as well as with both components. It is interesting that, the solutions for the scale factor do not depend on the gravitational constant but only on the 'time evolution' of the speed of light. GX-invariants act as efficient tools describing the models and the phases of the cosmological expansion.