Formal cubic-exponential two-by-two system ($\hat{Y}=\hat{G}{\lambda }^{\theta D}{e}^{({\lambda }^3/3+h\lambda )D}$)

For $D=\mathrm{diag}(1,-1)$, $A=D{\lambda }^2+\left(\begin{array}{cc}0& U\\ V& 0\end{array}\right)\lambda +\left(\begin{array}{cc}d& P\\ R& -d\end{array}\right)$, 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 isIts 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.