$\mathrm{Spin}(7)$-structures on complex linear bundles and explicit Riemannian metrics with holonomy group $\mathrm{SU}(4)$

A system of differential equations with 5 unknowns is fully investigated; this system is equivalent to the existence of a parallel $\mathrm{Spin}(7)$-structure on a cone over a 3-Sasakian manifold. A continuous one-parameter family of solutions to this system is explicitly constructed; it corresponds to metrics with a special holonomy group, $\mathrm{SU}(4)$, which generalize Calabi's metrics.
Bibliography: 10 titles.