Consider a circum-conic (C) with center T and perspector S = G/T (Ceva conjugate).
A variable line (L) passing through T meets (C) at two points M, N. Let C(M, N) be the bicevian conic passing through the vertices of the cevian triangles of M and N. (C) and (L) meet at two points P, Q and, when (L) rotates around T, the locus of P, Q is a cubic K(T).
K(T) is a pivotal cubic with pivot T and pole the barycentric square Ω of G/T. Hence, K(T) = K(G/S) = pK(S^2, G/S) = pK((G/T)^2, T).
The isoconjugation with pole S transforms K(T) into the isotomic pivotal cubic pK(G, aS) with pivot aS, the anticomplement of S.
Here is a selection of some cubics K(T) and K'(T) = pK(G, aS) with a list of centers X(i) when the cubic is not listed in CTC (updated by Peter Moses, 2021-12-21).
Remark : when T = G, the cubics K(T) and K'(T) are the union of the medians of ABC.