= Quadratic spectral Lax pair for zero-parameter Painlevé II
{title2=$w'=2u'w/u,\quad u''+u(w+z/2)=0$}
Let $A$ have diagonal entries $\pm(\lambda^2+w+z/2)$ and off-diagonal entries $u\lambda+u'$ and $-2w\lambda/u+2u'w/u^2$. Let $B$ have diagonal entries $\pm\lambda/2$ and off-diagonal entries $u/2,-w/u$. Substitution into the <isomonodromic deformation> compatibility identity gives the displayed equations on domains with $u\ne0$. Thus $(w/u^2)'=0$, so $w=Cu^2$ and $u''+Cu^3+zu/2=0$, a rescaled zero-parameter <Painlevé II equation> when $C\ne0$. The leading coefficient $-2w/u$ is required by cancellation of the quadratic spectral terms.
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