Non-expansive matrix number systems with bases similar to $J_n(1)$

We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is similar to Jn, the Jordan block of 1 of size n. If M = J2, we classify digit sets of size 2 allowing representation of the whole Z. For Jn with n ≥ 3, it is shown that three digits suffice to represent all of Z. For bases similar to Jn, at most n digits are required, with the exception of n = 1. Moreover, the language of strings representing the zero vector with M = J2 and the digits (0,±1) is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.