Probable structural mechanisms with structural invariants based on a chain of six-membered cycles linked through a common atom are proposed for some transitions between crystal water phases. It is shown that in the II → Ic transition with the structural invariant under consideration, 5/6 of the hydrogen bonds of the initial structure are retained, while in the same transition considered previously with the structural invariant in the form of a band of six-membered cycles linked through a shared bond, only 3/4 of the bonds of the initial structure are retained. In the V → Ic transition, both structural invariants (chains of six-membered cycles linked through a common atom and bands of six-membered cycles linked through a shared bond) are shown to be retained simultaneously. For the structural mechanism of the V → Ic transition with these structural invariants, 11/14 of hydrogen bonds of the initial structure are retained. It is shown that structural invariants of the mechanisms for the IX → Ic, VI → Ic, IX → II, and II → III transitions may also be based on a chain of six-membered cycles linked through a common atom.