On local automorphisms of certain quadrics of codimension 2

The article considers nondegenerate quadrics in $\mathbf{C}^{n+1}$ with codimension 2 that are of the form $M=\{z\in\mathbf{C}^n$, $\omega\in\mathbf{C}^2:\operatorname{Im}\omega_j=\langle z,z\rangle_j$; $j=1,2\}$, where $\langle z,z\rangle_j=\sum^n_{\mu,\nu=1^{\omega^j}\mu\nu^z\mu^{\bar{z}}\nu}$ are Hermitian forms, and thje stability groups $\operatorname{Aut}_xM$ that preserve the point $x$. It is proved that if the matrix $\omega^1$ is stable and the matrix $(\omega^1)^{-1}\omega^2$ has more than two different eigenvalues, all automorphisms of $\operatorname{Aut}_xM$ are linear transformations.