Inverse unsteady problems of high-speed hydromechanics were investigated with the use of integral-differential equation. Its solution was expressed as an asymptotic series and allowed obtaining analytic formulas for the first and second approximations both for the shape of slender axisymmetric cavities in ponderable and unponderable liquids for sub- and supersonic flows and for axisymmetric body shapes with the prescribed pressure distribution over the surface. Parameters limitations for the steady and the unsteady cavity flows were investigated with the use of stability principle for the mathematical physics problems. In the case of the partial cavitation on the conicalcylindrical bodies some physical effects were revealed; a classification of the possible axisymmetric cavity shapes was done. The gas ventilation influence on the slender axisymmetric steady cavities was investigated.