X(1), X(44), X(88), X(239), X(241), X(292), X(294), X(1931)
We meet this cubic in Clark Kimberling's TCCT p. 241. In the example above, Z+(IK) is the isogonal cK0 with root X(513) and node X(1). See also Table 69.
In the most general case, if L is a line, Z+(L) is the isogonal nK0 locus of point M such that the pole of the line MM* in the conic ABCMM* lies on L (M* is the isogonal conjugate of M). The root of Z+(L) is the isogonal conjugate of the trilinear pole of L.
The general equation of Z+(L) is : a^2 p x (c^2y^2 + b^2z^2) + cyclic = 0 where p x + q y + r z = 0 is the equation of L.
When L is the line at infinity, Z+(L) is Kjp and this is the only K60 of this type.
When L passes through O = X(3), Z+(L) is a focal cubic. See Z+(O).
If L is the trilinear polar of P = (p : q : r), Z+(L) is a K60 if and only if P lies on the sextic which is the isogonal transform of the K60 described here.