Analytical Solutions of Singular Isothermal Quadrupole Lens
  Using analytical method, we study the Singular Isothermal Quadrupole (SIQ)
lens system, which is the simplest lens model that can produce four images. In
this case, the radial mass distribution is in accord with the profile of the
Singular Isothermal Sphere (SIS) lens, and the tangential distribution is given
by adding a quadrupole on the monopole component. The basic properties of the
SIQ lens have been studied in this paper, including deflection potential,
deflection angle, magnification, critical curve, caustic, pseudo-caustic and
transition locus. Analytical solutions of the image positions and
magnifications for the source on axes are derived. As have been found, naked
cusps will appear when the relative intensity $k$ of quadrupole to monopole is
larger than 0.6. According to the magnification invariant theory of the SIQ
lens, the sum of the signed magnifications of the four images should be equal
to unity \citep{dal98}. However, if a source lies in the naked cusp, the summed
magnification of the left three images is smaller than the invariant 1. With
this simple lens system, we study the situations that a point source infinitely
approaches a cusp or a fold. The sum of magnifications of cusp image triplet is
usually not equal to 0, and it is usually positive for major cusp while
negative for minor cusp. Similarly, the sum of magnifications of fold image
pair is usually neither equal to 0. Nevertheless, the cusp and fold relations
are still equal to 0, in that the sum values are divided by infinite absolute
magnifications by definition.
