= Solution
To include the <stable-variable lag in nilpotent center-manifold reduction>, let $v=2w+h_3(z,w)+O(5)$. Invariance of the <center manifold> requires
$$
w\partial_z h_3-{2\over9}(z-2w)^3=-{3\over2}h_3.
$$
For the <homogeneous polynomial> $h_3=Az^3+Bz^2w+Czw^2+Dw^3$, comparison of coefficients successively gives
$$
A={4\over27},\quad B=-{32\over27},\quad C={272\over81},\quad D=-{832\over243}.
$$
Only $A$ is needed for the requested cubic <normal form>. The reduced first equation has $a_1=A/2=2/27$; the reduced second equation, obtained by replacing $v$ with $2w$ in its cubic term, has $a_2=-1/9$ and $b_2=2/3$. Therefore the <cubic elimination at a nilpotent double-zero point> yields
$$
\boxed{P=-{1\over9},\qquad Q={2\over3}+3{2\over27}={8\over9}.}
$$
In particular, ignoring the lag and imposing $v=2w$ exactly would incorrectly give $Q=2/3$. The full cubic graph above provides a direct substitution check.
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