  We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A)
\quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and
$G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the
affine group $\cG$, the group of affine transformations of the real axis
preserving the orientation. If $A$ is a symmetric operator, the group $G$
induces an action/flow on the operator unit ball of contractive transformations
from $\Ker (A^*-iI)$ to $\Ker (A^*+iI)$. We establish several fixed point
theorems for this flow. In the case of one-parameter continuous subgroups of
linear transformations, self-adjoint (maximal dissipative) operators associated
with the fixed points of the flow give rise to solutions of the (restricted)
generalized Weyl commutation relations. We show that in the dissipative
setting, the restricted Weyl relations admit a variety of non-unitarily
equivalent representations. In the case of deficiency indices $(1,1)$, our
general results can be strengthened to the level of an alternative.
