Modification of Poincaré's construction and its application in $CR$-geometry of hypersurfaces in $\mathbf{C}^4$

The modified Poincaré construction (a generalization of Poincaré's homological operator) was earlier used to estimate the dimension of the local automorphism group for an arbitrary germ of a real-analytic hypersurface in $\mathbf{C}^3$. In the present paper we prove the following alternative. For every hypersurface in $\mathbf{C}^4$, this dimension is either infinite or does not exceed $24$. Moreover, $24$ occurs only for a non-degenerate hyperquadric (one of the two). If the hypersurface is $2$-nondegenerate (resp. $3$-non-degenerate) at a generic point, the bound can be improved to $17$ (resp. $20$).