Variational problems in the gasdynamics of axisymmetric irrotational flows have been treated in a large number of papers up to the present time. The idea of considering a control contour, which appreciably simplifies the solution of the problems, was proposed by Nikol'skii in 1950 [1]. The method of solution of degenerate variational problems was worked out in 1946 by Okhotsimskii [2]. Guderley and Hantsche in 1955 [3] formulated the problem of the optimal supersonic nozzle are reduced it to a boundary problem for ordinary differential equations. In 1957 the author problems of gasdynamics of a perfect gas. The result of these papers, relating to axisymmetric nozzles, were repeated for the case of an imperfect gas by Rao in 1958 [5]. Rao's method differed from the method of [3,4], and its proof was given in 1959 by Guderley [6]. The cited papers touched on necessary conditions for an extremum, and [4] indicated a method of investigating the fulfillment of sufficient conditions. Fanselau in [7] returned to the study of sufficient conditions for an extremum, but did not obtain constructive results. Finally, Sternin in 1961 [8] derived the equation of the locus of points of extremal characteristics, at which the acceleration became infinite, and at the same time determined the region of applicability of the previously worked out solution with continuous functions. Here is developed a further study of variational problems of axisymmetric supersonic flows. Sufficient conditions are determined for attaining the minimum wave drag of bodies of revolution, discontinuous solutions are constructed for regions in which the minimum is not attained with continuous functions, and regions of isemtropic flow are delineated. The author is deeply grateful to O.S. Ryzhov for valuable discussion of the paper, and also to A.N. Belov, who carried out all the computations, and to L.V. Papadin for executing the graphs.