Here we introduce two pencils of cubics, each cubic is determined by a point on the Brocard Axis.
The first pencil is similar to, and indeed related to, the Euler Pencil given in Table 27 of CTC.
Each pencil contains a family of circumcevian invariant cubics, as also does the Euler Pencil.

The pencil of cubics of type pK appear also in the euler-brocard-jerabek net

k	P(k)	W(k)	φ	Q(φ),Q(-φ)	pK(X6,P(k))	pK(W(k),X3)	nK0(W(k),X6)	E(k)	P'(k)
∞	X30	X50	π/3	X15,X16	K001	K073	K148	X186	X74
1/3	X2	X32	0	X6,X6	K002	K172	nK0(X32,X6)	X25	X6
-1	X20	X577	π/2	X3,X3	K004	pK(X577,X3)	nK0(X577,X6)	X3	X3
1/2	X5	X2965	π/6	X61,X62	K005	K349	nK0(X2965,X6)	?	X1173
1	X4	X571	π/4	X371,X372	K006	pK(X571,X3)	nK0(X571,X6)	X24	X4
0	X3	X6		complex	K003	K003	K024	X4	X54
-5	?	?		X1151,X1152	K156				?

The Special Centres S(n)
These are triangle centres which arise in this theory, but are not yet in ETC.

n	first barycentric	notes
1	a2(2a2-b2-c2)/(b2-c2)	circumcevian invariant, note 1
2	a2(3a4+ 6b2c2 - (a2+b2+c2)2)/(b2-c2)	circumcevian invariant, note 2
3	a2(2b2c2-a2b2-a2c2)	note 3

Note 1
S(1) is the intersection of the Brocard Axis and the line X110X351.
The latter line is the tangent at X110 to the Circumcincle.
S(1) also lies on the circumconic with perspector X187, which passes through X6 and X110.
S(1) is circumcevian invariant, and lies on K148 = nK0(X50,X6). It is the third intersection
along with X15 and X16. Since K148 is circumcevian invariant, S(1) must be.

Note 2
S(2) is the inverse of S(1) in the Circumcircle.
S(2) is the intersection other than X3 of the Brocard Axis and the circle with diameter X3X110.
S(2) is circumcevian invariant as the inverse of S(1).

Note 3
S(3) is the intersection of the Lemoine Axis and the line X2X6.
S(3) is the perspector of the circumconic which passes through X6 and X99.
It is this circumconic which is important in the theory of the nK0(W,X6) above.
Each nK0 contains A,B,C, the Apollonian points, and three fixed points on the Circumconic.