  Analysis of the first-order corrections to higher-spin equations is extended
to homotopy operators involving shift parameters with respect to the spinor $Y$
variables, the argument of the higher-spin connection $\omega(Y)$ and the
argument of the higher-spin zero-form $C(Y)$. It is shown that a relaxed
uniform $(y+p)$-shift and a shift by the argument of $\omega(Y)$ respect the
proper form of the free higher-spin equations and constitute a one-parametric
class of vertices that contains those resulting from the conventional (no
shift) homotopy. A pure shift by the argument of $\omega(Y)$ is shown not to
affect the one-form higher-spin field $W$ in the first order and, hence, the
form of the respective vertices.
