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\begin{document}

\tit
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Арифметическая симметрия гамильтоновых систем, платоновы тела и интегрируемая
динамика тяжелого твердого тела с неподвижной точкой
}

\auth
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Д.~Л.~Абраров
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\addr
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Вычислительный центр РАН, Москва, Россия
}

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\noindent
На основе арифметической и геометрической интерпретации мероморфной
симметрии Ковалевской проводится классификация случаев аналитической
интегрируемой динамики тяжелого твердого тела с неподвижной точкой.
Приводится механическая и геометрическая интерпретация случаев 
интегрируемости.

Метод получения классификации основан на редукции универсальной группы
симметрии $PSL_2({\bf C})$ для систем указанного типа к симметриям минимальных
разрешимых групп Галуа --- классам сопряженных подгрупп знакопеременной
группы $A_4$. 

Механическая и геометрическая интерпретации основаны на возможности 
геометрической реализации группы $A_4$ и классов ее сопряженных подгрупп
в группе $SO(3)$ --- конфигурационном пространстве задачи о тяжелом твердом
теле с неподвижной точкой.

Работа выполнена при финансовой поддержке РФФИ (99-01-00785).  

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\item Абраров~Д.~Л. О симметриях гамильтоновых систем с $3/2$ степенями
свободы, порожденных полным набором интегралов.~// Задачи исследования
устойчивости и стабилизации движения. Часть 2. М.: Изд-во ВЦ РАН, 2000,
С~.3---28.
                
\item Абраров~Д.~Л. Гамильтоновы системы вертексного типа и классификация
случаев интегрируемой динамики тяжелого твердого тела c неподвижной
точкой.~// Задачи исследования устойчивости и стабилизации движения. М.:
Изд-во ВЦ РАН, 2001, в печати.

\end{refer}

\medskip

\tit
{
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Arithmetic Symmetry of Hamiltonian Systems, Plato's Bodies and Integrable
Dynamics of a Heavy Rigid Body Rotating about Fixed Point
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\auth
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D.~L.~Abrarov
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\addr
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Computing Center of the RAS, Moscow, Russia
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\noindent
The classification of analytic integrable cases in dynamics of a heavy rigid
body having fixed point is obtained. The method of the classification
is based on reduction (quantization) of the universal group symmetry
$PSL_2({\bf C})$ (Kovalevskaya meromorphic symmetry) of the considered
systems to simpliest Galois group symmetries isomorphic to the set of the
conjugacy classes  of the alternating group $A_4$.
 
The mechanical and geometric interpretations of the integrable motion are 
based on possibility of representation of the group $A_4$ in 
the group $SO(3)$ which is the configuration space of the problem. 

The work is supported by RFBR (99-01-00785).

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\item Abrarov,~D.~L. On Symmetries of Hamiltonian Systems with $ 3/2$ Degrees
of Freedom Generated by a Complete Set of First Integrals.~// Problems of
Stability and Stabilization of Motion. Moscow: Computing Center of the
Russian Academy of Science. 2000. Part 2. pp.~3---28.

\item Abrarov,~D.~L. Hamiltonian Systems of Vertex Type and the
Classification of Cases of Integrable Dynamics of a Heavy Rigid Body about a
Fixed Point.~// Problems of Stability and Stabilization of Motion. Moscow:
Computing Center of the Russian Academy of Science. 2001. To appear.

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