  Given a dynamical system $(A,\al)$ where $A$ is a unital $\ca$-algebra and
$\al$ is a (possibly non-unital) *-endomorphism of $A$, we examine families
$(\pi,\{T_i\})$ such that $\pi$ is a representation of $A$, $\{T_i\}$ is a
Toeplitz-Cuntz family and a covariance relation holds. We compute a variety of
non-selfadjoint operator algebras that depend on the choice of the covariance
relation, along with the smallest $\ca$-algebra they generate, namely the
$\ca$-envelope. We then relate each occurrence of the $\ca$-envelope to (a full
corner of) an appropriate twisted crossed product. We provide a counterexample
to show the extent of this variety. In the context of $\ca$-algebras, these
results can be interpreted as analogues of Stacey's famous result, for
non-automorphic systems and $n>1$.
  Our study involves also the one variable generalized crossed products of
Stacey and Exel. In particular, we refine a result that appears in the
pioneering paper of Exel on (what is now known as) Exel systems.
