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It follows from the above theorem that formulae (47) and 1–6 of the statement of
Theorem 3 give six inequivalent realizations of the Lie algebra of the Euclid group
E(4) having the basis elements Pµ = ?xµ and (44), (45). To get all possible realizations
of the algebra in question belonging to the above class it is necessary to add to the
list of realizations of the algebra AO(4) obtained in Theorem 3 the following three
?
realizations of the operators Ja , Ja :
+


J1 = ? sin u1 tan u2 ?u1 ? cos u1 ?u2 ,
+
1.
J2 = ? cos u1 tan u2 ?u1 + sin u1 ?u2 ,
+

?
J3 = ?u1 , Ja = 0;
+

J1 = ? sin u1 tan u2 ?u1 ? cos u1 ?u2 ? sin u1 sec u2 ?u3 ,
+
2.
J2 = ? cos u1 tan u2 ?u1 + sin u1 ?u2 ? cos u1 sec u2 ?u3 ,
+

?
J3 = ?u1 , Ja = 0;
+

?
Ja = 0, Ja = 0,
+
3.
where a = 1, 2, 3. This yields nine inequivalent realizations of the Lie algebra of the
group E(4).
In particular, the basis generators of the Euclid groups realized on the sets of
solutions of the Dirac and self-dual Yang–Mills equations in the Euclidean space R4
are reduced to such a form that the generators of the rotation groups are given by
(44), (45), Jµ? being adduced in the formulae 4 of the statement of Theorem 3.


6 Concluding remarks
Summarizing the results of Sections 3 and 4 yields the following structure of realizati-
ons of the Lie algebra of rotation group by LVFs in n variables:
• If n=1, then there are no realizations.
• As there is no realization of AO(3) by real non-zero 2 ? 2 matrices, the only
realization for the case n = 2 is given by (13). Furthermore, this realization is
essentially nonlinear (i.e., it is not equivalent to a realization of the form (9)).
• In the case n = 3 there are two more realizations (38) (which is equivalent to
(13)) and by formula (14). The latter realization is essentially nonlinear.
• Provided n > 3, there is no new realizations of AO(3) and, furthermore, any
realization can be reduced to a linear one (say, to (39)).
An evident (and very important) consequence of Theorem 1 is that there are only
two inequivalent classes of O(3)-invariant partial di?erential equations of order r.
On covariant realizations of the Euclid group 315

They are obtained via di?erential invariants of the order not higher than r of the Lie
algebras having the basis elements (13), (14). In particular, the Weyl, Maxwell, Dirac
equations are the special cases of the general system of ?rst-order partial di?erential
equations in n ? 8 dependent variables invariant with respect to the algebra (14). We
intend to devote one of our future publications to description of ?rst-order di?erential
invariants of the Lie algebra of the Euclid group E(3) having the basis elements (13),
(14) and (37). Let us note that this problem has been completely solved provided
basis elements of AE(3) are given by formulae (12) [20].
Acknowledgments. One of the authors (R.Zh.) gratefully acknowledges ?nancial
support from the Alexander von Humboldt Foundation.

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Contents
.I. , .. , ’i
ii i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
.I. , .. , i i
’i ii i i . . . . . . . . . . . . . . . . . . . . . . . 5
.I. , .. , i-iii i
–-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
.I. , .. , i ’
i i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
W.I. Fushchych, R.M. Cherniha, V.I. Chopyk, On unique symmetry
of two nonlinear generalizations of the Schr?dinger equation . . . . . . . . . . . . . . . . . .30
o
.I. , .I. , i iii i, iii
i i i-i . . . . . . . . . . . . . . . . . . . .36
W.I. Fushchych, P.V. Marko, R.Z. Zhdanov, Symmetry classi?cation
of multi-component scale-invariant wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
R.Z. Zhdanov, W.I. Fushchych, P.V. Marko, New scale-invariant nonlinear
di?erential equations for a complex scalar ?eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
P. Basarab-Horwath, L.L. Barannyk, W.I. Fushchych, Some exact solutions
of a conformally invariant nonlinear Schr?dinger equation . . . . . . . . . . . . . . . . . . . . 56
o
P. Basarab-Horwath, W.I. Fushchych, Implicit and parabolic ansatzes:
some new ansatzes for old equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
P. Basarab-Horwath, W.I. Fushchych, L.F. Barannyk, Solutions
of the relativistic nonlinear wave equation by solutions of the nonlinear
Schr?dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
o
P. Basarab-Horwath, W.I. Fushchych, O.V. Roman, On a new conformal
symmetry for a complex scalar ?eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
.I. , i i ii ii i . . . . 105
.I. , i i ? . . . . . . . . . . . . . . . . . . . 120
W.I. Fushchych, Symmetry of equations of nonlinear quantum mechanics . . . . . . .123
.I. , .. , i i i
i ii i i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
W.I. Fushchych, V.M. Boyko, Continuity equation in nonlinear quantum
mechanics and the Galilei relativity principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
W.I. Fushchych, A.G. Nikitin, Higher symmetries and exact solutions
of linear and nonlinear Schr?dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
o
.I. , .I. Ѻ, .. , i i
’– . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
W.I. Fushchych, Z.I. Symenoh, High-order equations of motion in quantum
mechanics and Galilean relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
W.I. Fushchych, Z.I. Symenoh, Symmetry of equations with convection terms . . 171
W.I. Fushchych, I.M. Tsyfra, On new Galilei- and Poincar?-invariant nonlinear e
equations for electromagnetic ?eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
V.I. Lahno, W.I. Fushchych, Reduction of self-dual Yang–Mills equations
with respect to subgroups of the extended Poincar? group . . . . . . . . . . . . . . . . . . .186
e
W.X. Ma, R.K. Bullough, P.J. Caudrey, W.I. Fushchych, Time-dependent
symmetries of variable-coe?cient evolution equations and graded
Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
R.Z. Zhdanov, W.I. Fushchych, On new representations of Galilei groups . . . . . . . 210
L.F. Barannyk, P. Basarab-Horwath, W.I. Fushchych, On the classi?cation
of subalgebras of the conformal algebra with respect to inner automorphisms 220
V.M. Boyko, W.I. Fushchych, Lowering of order and general solutions
of some classes of partial di?erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
W.I. Fushchych, What is the velocity of the electromagnetic ?eld? . . . . . . . . . . . . . .253
.I. , .I. , iii iii i
i i-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
.I. , .I. Ѻ, .. , i
ii ii i . . . . . . . . . . . . . . 266
.I. , .I. Ѻ, .. Ѻ, .. , i
i i i–i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
W.I. Fushchych, Z.I. Symenoh, I.M. Tsyfra, The Schr?dinger equation o
with variable potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych, Stationary mKdV hierarchy
and integrability of the Dirac equations by quadratures . . . . . . . . . . . . . . . . . . . . . 286
.I. , .I. Ѻ, .. , iii
i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych, On covariant realizations
of the Euclid group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

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