Scaling Invariance of the k[S]-Hierarchy and Its Strict Version

Let LTN(R)denote the algebra of NxN-matrices with coefficients from the commutative k-algebra R,k=R or C, that possess only afinite number of nonzero diagonals above the central diagonal. In a previous paper we discussed integrable deformations inside LTN(R)of various commutative subalgebras of LTN(k)that contain Sn, where S is theNxN-matrix corresponding to the shift operator. Here we focus on two deformations ofk[S], called the k[S]-hierarchy and itsstrict version and we discuss the scaling invariance that they possess. To do so, it is necessary to discuss both deformations from a wider perspective and consider them in a presetting instead of the usual setting. In this more general set-up we will present two k-subalgebras of R that are stable under the basic derivations of Rand such that these derivations commute on these k-subalgebras. This we apply at the introduction of the minimal realizations of both deformations, we show how these realizations relate to solutions in different settings and use them to show that both hierarchiespossess invariant scaling transformations.