One of the methods that allow to establish integrability and build solutions of differential equations of simplified models of mechanics and physics of continuous media is briefly described. The method is based on the close relationship between the integrability of equations and the analyticity of their solutions. Its occurrence is due to the Russian scientist Sofya Kovalevskaya and the French mathematician and politician Paul Penleve. Brief biographical information about them is given. The book contains an extended presentation of the materials of lectures delivered by the author to the participants of the All-Russian Scientific Youth School "Mechanics and its Applications in Technology and Technology," held in Moscow at the Institute of Problems of Mechanics of the Russian Academy of Sciences with the support of the Federal Target Program "Integration."