The problem of minimal realizations for \(n\)D linear systems is considered. Since general solutions do not exist in the general case, the aim of this paper is to emphasize a new class of systems for which this problem is solvable; this class consists of SISO \(n\)D linear systems represented by transfer functions of the form \[ f(s_{1},s_{2},\dots,s_{n})= \frac{a(s_{1},s_{2},\dots,s_{n})} {b(s_{1},s_{2},\dots,s_{n})} \] where \(a(s_{1},s_{2},\dots,s_{n})\) and \(b(s_{1},s_{2},\dots,s_{n})\) are multi-linear polynomials, i.e. they are of degree one in each indeterminate. In this case, the method proposed by the author in a previous paper (based on the construction of the companion matrix for the associated polynomial \(a_{f}(s_{1},s_{2},\dots,s_{n},s_{n+1})= s_{n+1}b(s_{1},s_{2},\dots,s_{n})-a(s_{1},s_{2},\dots,s_{n})\)) provides a set of equations which can be solved by isolating the independent ones. The existence and construction of minimal realizations are analyzed and a suitable procedure is given. If a minimal realization does not exist (i.e. the existence conditions do not hold), the so-called elementary operation algorithm developed by the author in a previous article is adapted to obtain a non-minimal realization, possibly of the least order. System equivalence is studied by employing variable transforms such as a generalized multivariable bilinear transform and variable inversion. Some examples illustrate the main ideas and the proposed procedures.