= First-order zero-energy solution in Grassmann supersymmetric mechanics
In the real <Grassmann-valued classical supersymmetric mechanics> variant, impose $\dot x=U(x)$ and $\psi_2=-\psi_1$. Then $\psi_1\psi_2=0$, and differentiating the first-order equation gives $\ddot x=UU'$. The remaining <Euler-Lagrange equations> reduce to $\dot\psi_1=-U'\psi_1$. These solutions have $E=0$ but $Q_1=2U\psi_1=-Q_2$, which need not vanish: $d(U\psi_1)/dt=U'U\psi_1-UU'\psi_1=0$.
For $U=1+cx$, one obtains $x(t)=e^{ct}x_0+(e^{ct}-1)/c$ and $\psi_1(t)=e^{-ct}\psi_{10}$ when $c\ne0$. At $c=0$, the limiting solution is $x(t)=x_0+t$, $\psi_1(t)=\psi_{10}$. In either case $Q_1=2(1+cx_0)\psi_{10}$.
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