Abstract
The Keldysh theorem for an elliptic equation with cha- racteristic parabolic degeneration is generalized for the case of an elliptic equation of the second-order canonical form with order and type degeneration. The criteria under which the Dirichlet or Keldysh problems are correct are given in a one-sided neighborhood of the degeneration segment, enabling one to write the criteria in a single form. Moreover, some cases are pointed out in which it is even ness- esary to give a criterion in the neighborhood because it is impossible to establish it on the segment of degeneracy of the equation.
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