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We consider the process of reaching the final state in the coevolving votermodel. There is a coevolution of state dynamics, where a node can copy a statefrom a random neighbor with probabilty $1-p$ and link dynamics, where a nodecan re-wire its link to another node of the same state with probability $p$.That exhibits an absorbing transition to a frozen phase above a critical valueof rewiring probability. Our analytical and numerical studies show that in theactive phase mean values of magnetization of nodes $n$ and links $m$ tend tothe same value that depends on initial conditions. In a similar way meandegrees of spins up and spins down become equal. The system obeys a specialstatistical conservation law since a linear combination of both typesmagnetizations averaged over many realizations starting from the same initialconditions is a constant of motion: $\Lambda\equiv (1-p)\mu m(t)+pn(t) =const$, where $\mu$ is the mean node degree. The final mean magnetization ofnodes and links in the active phase is proportional to $\Lambda$ while thefinal density of active links is a square function of $\Lambda$. If therewiring probability is above a critical value and the system separates intodisconnected domains, then the values of nodes and links magnetizations are notthe same and final mean degrees of spins up and spins down can be different.

Coupling of link- and node-ordering in the coevolving voter model