%\input amsppt.sty
%\documentstyle{amsppt}
%\nopagenumbers
\magnification=\magstep2
%\pageheight{26.5 true cm}
%\pagewidth{17 true cm}
%\vcorrection{-.5 true cm}
%\hcorrection{-.5 true cm}
\TagsOnRight
\pretitle{ \centerline{\bf { ����� ���� ����. ��⥬�⨪�. ��.~2}}
\centerline{\bf {�������, 1999}}

\noindent --------------------------------------------------------------------------------------------------
\vskip .1 cm
��� 514.83+514.7
\vskip 0.3 cm 
\centerline {\bf �.~�.~�좮�, �.~�.~��p����}
\vskip .3 cm
%\leftheadtext{\it �.~�.~�좮�, �.~�.~��p����}
%\rightheadtext{\it �����p�������� ����}
\topmatter
%\author �.�.�좮�, �.�.��p����
%\endauthor
\title
\centerline{\bf �����p�������� ����, ����p�����} 
\centerline{\bf �⭮�⥫쭮 ��p������᪮�� ����}
%\footnote"*"{}
\endtitle
\abstract \nofrills
\vskip .3 cm\noindent
����祭 ����� �����p��������� �����, ����p������ �⭮�⥫쭮
��p������᪮�� ���� � ����� ��⭮� ��砥.

%�������p���: 1 ����.
\endabstract
\endtopmatter
%\footnote""{* ����� �믮����� �p� 䨭��ᮢ�� �����p���
%������p�⢠ ��饣� �
% �p���ᨮ���쭮�� ��p�������� �� (�p���� 97-0-1.3-99). }
\document
\pageno=1
\vskip 0.5 cm
\def\1{\partial}
\def\2{\lambda}
\def\4{\tilde}
\def\5{\frac}
\def\6{\sqrt{1-(\4x^4)^2}}
\def\7{{ {2(\4x^1)^2+2\4x^4-(\4x^1\4x^2)^2} \over {2\4x^1} }}
\def\8{{ {\4x^2} \over {\4x^1} }}
\def\9{{ \4x^1\4x^2 }}
\def\3{{ {2\4x^4-(\4x^1\4x^2)^2} \over {2\4x^1} }}
��宦����� ����ᮢ �����p��������� �����, ����᪠��� �p㯯� ᨬ���p��
����� ������ ���祭�� ��� ����祭�� ��p��� ��⥣p���� �p������� ��p��� 
(��p���� �.�. ��p�� ��⥣p��� �p�������
��p��� (������ p�����⮢)//����. �p. ����. ���. �-�. ��⥬�⨪�.
��.1. �������,
 1997.  C.101 - 110).
� �����饩 p���� ����祭 ����� �����p��������� �����, ����p������
�⭮�⥫쭮 ��p������᪮�� ���� � ����� ��⭮� ��砥.

����� � �p���p���⢥ $\Bbb R^4_3$ ������ ����
$$
\xi^i=(-x^2-x^4, x^1+\2, \mu, -x^1). \eqno(1)
$$
���⢥������� �p㯯� ���䥮��p䨧��� $G_1$ ����� ���
$$
\4 x^1=\2{{a^2}\over 2}-a(x^2+x^4)+x^1,\ \
\4 x^2=\2{{a^3}\over 3}-{{a^2}\over 2}(x^2+x^4)+a(x^1+\2)+x^2,
$$
$$
\4 x^3=\mu a+x^3,\ \
\4 x^4=-\2{{a^3}\over 3}+{{a^2}\over 2}(x^2+x^4)-x^1a+x^4, \eqno(2)
$$
��� $a$ -- �p㯯���� ��p����p.

��� p�襭�� ����� ���� ���� ���ᨬ���p��� ⥭��p $F_{ij}$,
㤮���⢮p��騩 �p������� ���ᢥ���
$$
\1_{[i}F_{jk]}=0 \eqno(3)
$$
� �᫮��� ����p���⭮��
$$
L_{\xi}F_{ij}=\xi^s \1_s F_{ij}+F_{sj}\1_i \xi^s+F_{is}\1_j\xi^s=0 \eqno(4)
$$
( $L_{\xi}$ --- �p��������� ��).

����ᠢ  �p������� (4) �p� $\2=\mu=0$ ��� p������� ���祭�� �����ᮢ, ����稬
᫥������ ��⥬�:
$$
\align
&-(x^2+x^4)\1_1F_{12}+x^1\1_2F_{12}-x^1\1_4F_{12}+F_{24}=0, \eqno(5.1) $$ $$
&-(x^2+x^4)\1_1F_{13}+x^1\1_2F_{13}-x^1\1_4F_{13}+F_{23}+F_{34}=0,\eqno(5.2) $$ $$
&-(x^2+x^4)\1_1F_{14}+x^1\1_2F_{14}-x^1\1_4F_{14}+F_{24}=0,\eqno(5.3) $$ $$
&-(x^2+x^4)\1_1F_{23}+x^1\1_2F_{23}-x^1\1_4F_{23}-F_{13}=0,\eqno(5.4) $$ $$
&-(x^2+x^4)\1_1F_{24}+x^1\1_2F_{24}-x^1\1_4F_{24}+F_{12}-F_{14}=0,\eqno(5.5) $$ $$
&-(x^2+x^4)\1_1F_{34}+x^1\1_2F_{34}-x^1\1_4F_{34}+F_{13}=0,\eqno(5.6)
\endalign
$$
��� p�襭�� �⮩ ��⥬� ᤥ���� ������ ���p�����
$$
\4x^1=x^2+x^4,\ \4 x^2={{x^1}\over {x^2+x^4}},
\ \4x^3=x^3,\ \4 x^4={{1}\over 2}(x^1)^2+x^2(x^2+x^4). \eqno(6)
$$
��᫥ ⠪�� ������ ��p������� ��⥬� (5) �p���� ���:
$$
\align
&{ {\1 F_{12}}\over {\1\4x^2} } +F_{24}=0,  \eqno(7.1) $$ $$
&{ {\1 F_{13}}\over {\1\4x^2} } +F_{23}+F_{34}=0,  \eqno(7.2) $$ $$
&{ {\1 F_{14}}\over {\1\4x^2} } +F_{24}=0,  \eqno(7.3) $$ $$
&{ {\1 F_{23}}\over {\1\4x^2} } -F_{13}=0,  \eqno(7.4) $$ $$
&{ {\1 F_{24}}\over {\1\4x^2} } +F_{12}-F_{14}=0,  \eqno(7.5) $$ $$
&{ {\1 F_{34}}\over {\1\4x^2} }+F_{13}=0.  \eqno(7.6)
\endalign
$$

���� ��⥬� �p������� (7.1),(7.3),(7.5) � (7.2),(7.4),(7.6), ����稬
᫥���騥 ��p������ ��� ��������� ⥭��p� $F_{ij}$:
\pagebreak
$$
\align
&F_{13}=\4x^2C_1+C_2,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
F_{24}=\4x^2C_5+C_6, $$ $$
&F_{23}={{(\4x^2)^2}\over 2} C_1+\4x^2C_2+C_3,\ \ \ \
F_{12}=-{{(\4x^2)^2}\over 2} C_5-\4x^2C_6+C_7, \eqno(8) $$ $$
&F_{34}=-{{(\4x^2)^2}\over 2} C_1-\4x^2C_2+C_4,\ \
F_{14}=-{{(\4x^2)^2}\over 2} C_5-\4x^2C_6+C_8,
\endalign
$$
��� $C_i=C_i(\4x^1, \4x^3, \4x^4),\ i=1,\,\dots,\,8.$

����⠢��� ���������� (8) � ��⥬� �p������� ���ᢥ��� (3), ����稬 ��᫥
�������p��� �p���p��������
$$
\align
&{1\over {\4 x^1}} (\4x^2C_1+C_2)+\9\left( {{(\4x^2)^2}\over 2} 
 { {\1C_1}\over {\1\4x^4} }+
 \4x^2 {{\1C_2}\over {\1\4x^4}} + {{\1C_3}\over {\1\4x^4}} \right)- 
$$ $$
&-\left( \4x^2 {{\1C_1}\over {\1\4x^1}} + {{\1C_2}\over {\1\4x^1}} - 
  {{\4x^2}\over {\4x^1}} C_1 +
  \7 (\4x^2 {{\1C_1}\over {\1\4x^4}} + {{\1C_2}\over {\1\4x^4}})  
  \right)- 
$$ $$
&- {{(\4x^2)^2}\over 2} {{\1C_5}\over {\1\4x^3}} - 
  \4x^2 {{\1C_6}\over {\1\4x^3}} + {{\1C_7}\over {\1\4x^3}}
=0, \eqno(9.1)
$$ $$
& {{C_5}\over {\4x^1}} + \9\left( \4x^2 {{\1C_5}\over {\1\4x^4}} + 
  {{\1C_6}\over {\1\4x^4}} \right)-
  {{\1C_8}\over {\1\4x^1}} - 
$$ $$
&-\7\left(- {{(\4x^2)^2}\over 2} {{\1C_5}\over {\1\4x^4}} -
  \4x^2 {{\1C_6}\over {\1\4x^4}} + {{\1C_8}\over {\1\4x^4}} \right) + 
  {{\1C_7}\over {\1\4x^1}} + 
$$ $$
&+\3\left(- {{(\4x^2)^2}\over 2} {{\1C_5}\over {\1\4x^4}} -
  \4x^2 {{\1C_6}\over {\1\4x^4}} + {{\1C_7}\over {\1\4x^4}} 
  \right)=0, \eqno(9.2)
$$ $$
& {1\over {\4x^1}} (-\4x^2C_1-C_2) + \9\left(- {{(\4x^2)^2}\over 2} 
  {{\1C_1}\over {\1\4x^4}} -
  \4x^2 {{\1C_2}\over {\1\4x^4}} + {{\1C_4}\over {\1\4x^4}} \right) +
$$ $$
&+ {{(\4x^2)^2}\over 2} {{\1C_5}\over {\1\4x^3}} + \4x^2\left(
  {{\1C_1}\over {\1\4x^1}} -
  {{\1C_6}\over {\1\4x^3}} \right) + {{\1C_8}\over {\1\4x^3}} + 
  {{\1C_2}\over {\1\4x^1}} - 
$$ $$
&-\8C_1+\3\left(\4x^2 {{\1C_1}\over {\1\4x^4}} + {{\1C_2}\over {\1\4x^4}}
\right)=0, \eqno(9.3)
$$ $$
& {{\1C_4}\over {\1\4x^1}} + \7\left(- {{(\4x^2)^2}\over 2} 
  {{\1C_1}\over {\1\4x^4}} -
  \4x^2 {{\1C_2}\over {\1\4x^4}} + {{\1C_4}\over {\1\4x^4}} \right) -
$$ $$
&-\4x^2 {{\1C_5}\over {\1\4x^3}} - {{\1C_6}\over {\1\4x^3}} + 
  {{\1C_3}\over {\1\4x^1}} +
$$ $$
&+\3\left( {{(\4x^2)^2}\over 2} {{\1�_1}\over {\1\4x^4}} + 
  \4x^2 {{\1C_2}\over {\1\4x^4}} +
  {{\1C_3}\over {\1\4x^4}} \right)=0. \eqno(9.4)
\endalign
$$

���뢠� �������� ������ᨬ���� �㭪権 $1, \ \4x^2, \ (\4x^2)^2,
\ (\4x^2)^3$, $(\4x^2)^4,$
����稬 �� (9.1):
$$
\align
{{\1C_1}\over {\1\4x^4}}=0, \eqno(10.1) 
$$ $$
3\4x^1{{\1C_2}\over {\1\4x^4}} - {{\1C_5}\over {\1\4x^3}}=0, \eqno(10.2) 
$$ $$
{2\over {\4x^1}} C_1+\4x^1 {{\1C_3}\over {\1\4x^4}} - {{\1C_1}\over {\1\4x^1}}-
{{\1C_6}\over {\1\4x^3}} =0, \eqno(10.3) 
$$ $$
{1\over {\4x^1}} C_2- {{\1C_2}\over {\1\4x^1}} - {{(\4x^1)^2+\4x^4}\over {\4x^1}}
{{\1C_2}\over {\1\4x^4}} + {{\1C_7}\over {\1\4x^3}} =0; \eqno(10.4)
\endalign
$$
�� (9.2):
$$
\align
{{3}\over 2} \4x^1 {{\1C_5}\over {\1\4x^4}} + {{\4x^1}\over 2} 
{{\1C_8}\over {\1\4x^4}} -
{{\4x^1}\over 2} {{\1C_7}\over {\1\4x^4}} &=0, \eqno(10.5)
$$ $$
{{\1C_6}\over {\1\4x^4}} &=0, \eqno(10.6)
$$ $$
{1\over {\4x^1}} C_5- {{\1C_8}\over {\1\4x^1}} - {{(\4x^1)^2+\4x^4}\over {\4x^1}}
{{\1C_8}\over {\1\4x^4}} + {{\1C_7}\over {\1\4x^1}} + {{\4x^4}\over {\4x^1}}
{{\1C_7}\over {\1\4x^4}} &=0; \eqno(10.7)
\endalign
$$
�� (9.3):
$$
\align
-3\4x^1 {{\1C_2}\over {\1\4x^4}} + {{\1C_5}\over {\1\4x^3}} &=0, \eqno(10.8) 
$$ $$
-{2\over {\4x^1}} C_1+\4x^1 {{\1C_4}\over {\1\4x^4}} + {{\1C_6}\over {\1\4x^3}}+
{{\1C_1}\over {\1\4x^1}} &=0, \eqno(10.9) 
$$ $$
-{1\over {\4x^1}} C_2- {{\1C_8}\over {\1\4x^3}} + {{\1C_2}\over {\1\4x^1}} +
{{\4x^4}\over {\4x^1}} {{\1C_2}\over {\1\4x^4}} &=0; \eqno(10.10) 
\endalign
$$
�� (9.4):
$$
\align
{{\1C_4}\over {\1\4x^4}} - {{\1C_3}\over {\1\4x^4}} &=0, \eqno(10.11) 
$$ $$
-\4x^1 {{\1C_2}\over {\1\4x^4}} - {{\1C_5}\over {\1\4x^3}} &=0, \eqno(10.12)
$$ $$
{{\1C_1}\over {\1\4x^1}} + {{(\4x^2)^2+\4x^4}\over {\4x^1}} {{\1C_4}\over {\1\4x^4}} -
{{\1C_6}\over {\1\4x^3}} + {{\1C_3}\over {\1\4x^1}} + {{\4x^4}\over {\4x^1}} 
{{\1C_3}\over {\1\4x^4}} &=0.
\eqno(10.13)
\endalign
$$

�� (10.1) ᫥���, �� $C_1$ �� ������ �� $\4x^4$, �.�.
$C_1=C_1(\4x^1, \4x^3)$; �� (10.6) ��⥪���, �� $C_6=C_6(\4x^1, \4x^3)$.
�����뢠� �ࠢ����� (10.2) � (10.8), ����砥�, �� $C_2=C_2(\4x^1, \4x^3),
C_5=C_5(\4x^1, \4x^4)$.
�����뢠� �ࠢ����� (10.4) � (10.10), ����稬
$$
{{\1C_7}\over {\1\4x^3}}-{{\1C_8}\over {\1\4x^3}}=0. \eqno(11.1)
$$
�㬬� �ࠢ����� (10.3) � (10.9) ࠢ��ᨫ쭠 �ࠢ����� (10.11).

����� ��ࠧ��, ��騩 ��� �����஬����⭮�� ����, ����ਠ�⭮��
�⭮�⥫쭮 ��ࠡ����᪮�� ���� � $\xi^i=(-x^2-x^4, x^1+\2, \mu, -x^1)$
�� $\2=\mu=0$ �������� ��⥬�� ࠢ���� (8), ��� �㭪樨
$C_i=C_i(\4x^1, \4x^3, \4x^4),$ $i=3,4,7,8, \ C_j=C_j(\4x^1, \4x^3), j=1,2,6$, 
�
$C_5=C_5(\4x^1, \4x^4)$
㤮���⢮���� �ࠢ����� (10.3), (10.5), (10.7), (10.11), (11.1) �
$$
{1\over {\4x^1}}C_2-{{\1C_2}\over {\1\4x^1}}+{{\1C_7}\over {\1\4x^3}}=0, \eqno(11.2)
$$
$$
{{\1C_1}\over {\1\4x^1}}+{{(\4x^2)^2+2\4x^4}\over {\4x^1}} {{\1C_4}\over {\1\4x^4}}-
{{\1C_6}\over {\1\4x^3}} + {{\1C_3}\over {\1\4x^1}} =0. \eqno(11.3)
$$

\end
