@Carl Love 

First of all: thanks for many useful hints. Since parameter k is dependent from other numbers it seemed reasonable for me to look for recursive method. In the meantime I had the following idea:
Let k and S[1],...,S[k] be given as in my question. Put
R[0]:=[];
R[i]:=[op(R[i-1]),S[i]];
od;
U:=cartprod(R[k]);
while not U[finished] do ....
etc...

@Kitonum Since I'm working in finite rings Zn (c.f. Love) this is an elegant method, which can also be used in the more general situation where the module n is a product of primes, each congruent to 1 mod 4.

@Carl Love 

Hello, its really miraculous. Today I started a new MAPLE worksheet and added your proposed lines - and now it works. Thank you very much for your help, Gerd

@Carl Love 

When starting this procedure Orbit I get the error response

Error, reserved word `do` unexpected

Gerd

@Carl Love 

Hello, I've made a few "micro changes" concerning ":", ";" and now your program runs beautifully. Thank you!, Gerd

@Carl Love 

Thank you very much, unfortunately, however, it does not work with my MAPLE 18.2. Do you use a much more younger MAPLE version?
Indeed I'm working on Pell equations, where orbits, in my opinion, play an important role.

@tomleslie 

Hello; thank you very much for your idea to use array. It works!