The Cayley Cubic together with its parabolic curve (i.e. its intersection with its Hessian): cayley_parabolic_curve.sux

The so-called swallowtail. Its real locus contains a one-dimensional part (which is part of its singular locus).
Such lower-dimensional parts will not be shown automatically by surfex; however, this can be done if you call surfex via the computer algebra system Singular. We implemented a Singular library called surfex.lib for this purpose. Here is the sux-file for the image at the left: swallowtail_singular.sux