Subalgebras of the Lie algebra AC(2, 2) of the group C(2, 2), which is the group of conformal transformations of the pseudo-Euclidean space R 2,2 , are studied. All subalgebras of the algebra AC(2, 2) are splitted into three classes, each of those is characterized by the isotropic rank 0, 1, or 3. We present the complete classification of the class 0 subalgebras and also of the class 3 subalgebras which satisfy an additional condition. The results obtained are applied to the reduction problem

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... r the d'Alembert equation 2u + λu 3 = 0 in the space R 2,2 . J 46 , J 56 7 (+ + −)(+ − −) J 12 , J 14 , J 24 ⊕ J 35 , J 36 , J 56 8 (+)(+)(+ − −−) AO(1, 3) = J ab | a, b = 3, . . . 6 9 (+ + +−)(−)(−) AO(3, 1) = J ab a, b = 1, . . . , 4 10 (+)(++)(− − −) J 23 ⊕ J 45 , J 46 , J 56 11 (+)(+ + −)(−−) J 23 , J 24 , J 34 ⊕ J 56 12 (++)(+ − −)(−) J 12 ⊕ J 34 , J 35 , J 45 13 (+ + +)(−−)(−) J 12 , J 13 , J 23 ⊕ J 45 14 (+)(+)(+)(− − −) AO(3) = J 45 , J 46 , J 56 15 (−)(−)(−)(+ + +) AO(3) = J 12 , J 13 , J 23 Class 3 subalgebras of the algebra AO(3, 3). In the present paragraph, the problem of classification of class 3 subalgebras L ⊂ AO(3, 3) is reduced to the problem of classification of subalgebras of the algebra AIG(3, R), which is the Lie algebra of the group of nonuniform real transformations of the three-dimensional real space. Let L ⊂ AO(3, 3) be an arbitrary class 3 subalgebra. By virtue of the Witt theorem, we can assume that L leaves a subspace V (3) = Q 1 + Q 4 , Q 2 + q 5 , Q 3 + Q 6 invariant. All such subalgebras are contained in the maximal class 3 subalgebra A (3) which is a normalizer in AO(3, 3) of the totally isotropic space V (3) . According to [3] , every element J of the algebra A (3) can be uniquely represented in the form