Completely regular codes in the infinite hexagonal grid

A set $C$ of vertices of a simple graph is called a completely regular code if for each $i=0$, $1$, $2, \ldots$ and $j = i-1$, $i$, $i+1$, all vertices at distance $i$ from $C$ have the same number $s_{ij}$ of neighbors at distance $j$ from $C$. We characterize the completely regular codes in the infinite hexagonal grid graph.