The present article continues the author's cycle of papers concerning the local geometry of a CR-manifold [Math. Notes 47, No. 3, 239-242 (1990); translation from Mat. Zametki 47, No. 3, 17-22 (1990; Zbl 0715.32011); Math. Notes 48, No. 2, 721-725 (1990); translation from Mat. Zametki 48, No. 2, 3-9 (1990; Zbl 0718.32010), and Math. USSR, Sb. 72, No. 1, 189-205 (1992); translation from Mat. Sb. 182, No. 2, 203-219 (1991; Zbl 0776.32018)]. The action of a holomorphic transformation on the Levi form \(\langle z,z \rangle\) reduces to the action of \(\text{GL} (n, \mathbb{C}) \times \text{GL} (k, \mathbb{R})\) according to the formula \(\langle z,z \rangle \to \rho \langle C^{-1} z,\;C^{-1} z \rangle\). The invertible linear mappings of the form \(z \to Cz\), \(w \to \rho w\), satisfying the condition \(\langle Cz, Cz \rangle = \rho \langle z,z \rangle\), is the subgroup of linear automorphisms of a given quadric. Suppose that there is a collection of \(k\) matrices \(H^1, \ldots, H^k\). The author's aim is to determine and investigate the invariants of the above-mentioned action \[ \Gamma (p) = \{z \in \mathbb{C}^n : \text{rank} (H^1z, \ldots, H^kz) \leq p\}, \] \[ \gamma (q) = \{w \in \mathbb{C}^k : \text{rank} (w^1H^1 + \cdots + w^kH^k) \leq q\}. \] These invariants can be viewed in the context of generalizing the problem of eigenvalues and eigenvectors. Some of the properties of the invariants just introduced are listed. In order to give an idea of the distribution of dimensions of \(\Gamma\) and \(\gamma \) for a quadric in general position in various cases, the data for lower dimensions and codimensions are presented.