Shifted homotopy analysis of the linearized higher-spin equations in arbitrary higher-spin background

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