Geodesic
Spectral properties of operators with polynomial invariants in real finite-dimensional spaces
Informatics and Automation, Differential equations and topology. I, Tome 268 (2010), pp. 155-167

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We consider linear operators lying in the orthogonal group of a quadratic form and study those spectral properties of such operators that can be expressed in terms of the signature of this form. We show that in the typical case these transformations are symplectic. Some of the results can be extended to the general case when the operator admits a homogeneous form of degree $\ge3$. 
@article{TRSPY_2010_268_a11,
     author = {V. V. Kozlov},
     title = {Spectral properties of operators with polynomial invariants in real finite-dimensional spaces},
     journal = {Informatics and Automation},
     pages = {155--167},
     publisher = {mathdoc},
     volume = {268},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_268_a11/}
}
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V. V. Kozlov. Spectral properties of operators with polynomial invariants in real finite-dimensional spaces. Informatics and Automation, Differential equations and topology. I, Tome 268 (2010), pp. 155-167. http://geodesic.mathdoc.fr/item/TRSPY_2010_268_a11/