  The static kink, sphaleron and kink chain solutions for a single scalar field
$\phi$ in one spatial dimension are reconsidered. By integration of the
Euler--Lagrange equation, or through the Bogomolny argument, one finds that
each of these solutions obeys a first-order field equation, an autonomous ODE
that can always be formally integrated. We distinguish the BPS case, where the
required integral is along a contour in the $\phi$-plane, from the semi-BPS
case, where the integral is along a contour in the Riemann surface
double-covering the $\phi$-plane, and is generally more complicated.
