In the paper the authors first introduce a euclidon solution of the Einstein equations. This is a solution to the Einstein vacuum equations which is axially symmetric and stationary and flat. Physically, a solution of this kind characterizes some relativistic noninertial frame of reference in flat space-time. Then the authors use this euclidon solution to generate new axially symmetric and stationary solutions to the vacuum Einstein equations. par The method of generating these new solutions is based on a nonlinear composition of an arbitrary axially symmetric and stationary solution to the vacuum Einstein equations called the seed solution with the euclidon solution. par Finally, the authors show that the method reduces to integrating a set of first-order linear differential equations. Any set of equations of this kind can be solved in general because methods of treating a set of first-order linear differential equations are very well known. par The paper is interesting and important for the problem of obtaining new exact solutions to the Einstein equations.