  In the present paper we continue the project of systematic construction of
invariant differential operators on the example of the non-compact algebra
$G_{2(2)}$. We use both the minimal and the maximal Heisenberg parabolic
subalgebras. We give the main multiplets of indecomposable elementary
representations. This includes the explicit parametrization of the invariant
differential operators between the ERs. These are new results applicable in all
cases when one would like to use $G_{2(2)}$ invariant differential operators.
