The smallest Mealy automaton generating an indicable regular branch group

We construct a two-state Mealy automaton A over the three-letter alphabet generating a regular branch group G(A), which surjects onto the infinite cyclic group. Some algebraic and geometric properties of the group G(A) are derived. In particular, this group has a nearly finitary subgroup of index two, is amenable, just non-solvable, has exponential growth, and its action on the corresponding regular rooted tree is self-replicating, contracting, but it does not have the congruence subgroup property. We also derive in detail an ascending finite L-presentation for the group G(A).