In generalized optical systems, that is, in systems which may contain thin refracting elements of asymmetric dioptric power, pencils of rays may exhibit phenomena that cannot occur in conventional optical systems.  In conventional optical systems astigmatic pencils have two principal meridians that are necessarily orthogonal; in generalized systems the principal meridians can be at any angle.  In fact in generalized systems a pencil may have only one principal meridian or even none at all.  In contrast to the line foci in the conventional interval of Sturm line foci in generalized systems may be at any angle and there may be only one line focus or no line foci.  A conventional cylindrical pencil has a single line focus at a finite distance but it can be regarded as having a second line focus at infinity.  Only in generalized systems is a single line focus possible without a second at infinity or anywhere else.  The purpose of this paper is to illustrate the types of pencils possible in generalized systems.  Particular attention is paid to the effect of including an antisymmetric component in the divergency of the pencil.