---
abstract: |
    For the integrable system on $e(3,2)$ found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.

    MSC 70E05,70E17,37J15, 37J20
author:
- 'M. P. Kharlamov'
date: '05.09.2013'
title: |
    Extensions of the Appelrot classes\
    for the generalized gyrostat in a double force field
---

Introduction {#secc}
============

In the beginning of the 20th century G.G.Appelrot (H.Appelroth), in a series of his publications, gave deep and profound investigations of the motions of the Kowalevski top [@Kowa]. In [@App3; @App4], he found some specific families of motions and called them [*especially remarkable motions*]{}. These results were then included into Appelrot’s large treatise on the gyroscopes theory and published under common title as [@Appel]. Appelrot 