The work deals with the questions of the solvability of the Cauchy type functional equations introduced recently and proved to be not only very much interesting objects of functional analysis but also an important technical tool in solving quite different problems in such diverse fields as integral geometry, partial differential equations, the measure theory, ergodic theory, functional equations. The major part of the results is obtained on the base of the new dynamical method which has been worked out especially for studying functional equations in question. Two of the possible applications in Analysis of the results obtained (in Integral geometry and in the theory of hyperbolic differential equations) are discussed in the final part of the paper.