  R.Pavlov and S.Schmieding provided recently some synthetic results about
generic $\mathbb{Z}$-shifts, which rely mainly on an original theorem stating
that isolated points form a residual set in the space of $\mathbb{Z}$-shifts
such that all other residual set must contain it. As a direction for further
research, they pointed towards genericity in the space of $\mathbb{G}$-shifts,
where $\mathbb{G}$ is a finitely generated group. In the present text, we
approach this for the case of $\mathbb{Z}^d$-shifts, where $d \ge 2$. As it is
usual, multidimensional dynamical systems are much more difficult to
understand. Provided the result of R.Pavlov and S.Schmieding, it is natural to
begin with a better understanding of isolated points. We prove here a
characterization of such points in the space of $\mathbb{Z}^d$-shifts, in terms
of the natural notion of maximal subsystems which we also introduce in this
article. From this characterization we recover the one of R.Pavlov and
S.Schmieding's. We also prove a series of results which exploit this notion. In
particular some transitivity-like properties can be related to the number of
maximal subsystems. Furthermore, on the contrary of dimension one, the set of
isolated shifts is not residual. We also prove that the Cantor-Bendixson rank
of the space of $\mathbb{Z}^d$-shifts is infinite when $d > 1$, while it is
equal to one when $d=1$.
