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The paper is devoted to studying the method of the Sonin and Poisson transmutation operators for ordinary differential operators with variable coefficients. A new class of singular pseudodifferential operators is introduced which is closed under multiplication and which contains differential operators with singular coefficients, in particular, the Bessel operator. It is shown that after application of transmutation operators the original operator with variable coefficients (even a differential operator) is transformed into a classical pseudodifferential operator. The converse is also true: a pseudodifferential operator is transformed into a singular pseudodifferential operator. In all cases formulas are given for computing the symbols.Bibliography: 6 titles.

The Cauchy problem is considered for equations which decompose into Euler-Poisson-Darboux factors. A representation of the solution of this problem is obtained in terms of spherical means of the initial function. This makes it possible both to establish precise conditions for classical solvability and to investigate the question of the presence of lacunae. We note that a special feature of this class of problems is the appearance of lacunae in even-dimensional space and the appreciable effect of the parameter in the Bessel operator on the smoothness of the initial data and the order of the lacunae obtained. Bibliography: 11 titles.