Symmetry Supporting a Transition to Zero Cosmological Constant State

In a number of previous publications we demonstrated that the Two Measures Field Theory (TMT) enables to resolve the old cosmological constant (CC) problem avoiding the Weinberg's no-go CC theorem and together with this TMT agrees with all tests of the Einstein's general relativity and allows inflationary scenarios. Analysis performed in the present paper shows that there exists an intrinsic symmetry of TMT which emerges in the $\Lambda =0$ ground state. This symmetry contains a subgroup of reflections of the metric $g_{\mu\nu}\to -g_{\mu\nu}$ studied recently by a number of authors as the symmetry imposing zero CC. We show that realization of this idea in TMT is free of fine tuning and has no problems typical to other approaches.