= Formal cubic-exponential two-by-two system
{title2=$\widehat Y=\widehat G\lambda^{\theta D}e^{(\lambda^3/3+h\lambda)D}$}
For $D=\operatorname{diag}(1,-1)$, $A=D\lambda^2+\left(\begin{smallmatrix}0&U\\V&0\end{smallmatrix}\right)\lambda+\left(\begin{smallmatrix}d&P\\R&-d\end{smallmatrix}\right)$, the formal diagonal exponent has $h=d+UV/2$ and $\theta=(UR+VP)/2$. Matching the gauge equation gives the off-diagonal first coefficient $(-U/2,V/2)$ and the diagonal one $\rho D$, where $\rho=UV(d+h)/4-PR/2$. The second coefficient has diagonal entries $\rho^2/2-UV/8\pm h\theta/2$ and off-diagonal entries $(U\rho-P)/2,(V\rho+R)/2$. With $G_0=I$ and negative-index coefficients zero, the coefficient comparison is
$$
[D,G_m]=-A_1G_{m-1}-A_2G_{m-2}+G_{m-2}hD+G_{m-3}[\theta D-(m-3)I].
$$
Its diagonal equations at $m=2,3,4,5$ give $h,\theta,\rho$ and the second diagonal coefficient respectively. Its off-diagonal equations determine the next coefficients because the <commutator> with $D$ multiplies the upper and lower entries by $2$ and $-2$. This proves the expansion and supplies the recursion for all higher orders. The generally nonzero $\theta$ supplies <formal monodromy>.
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