Summary: A new necessary condition for adaption of elastic plastic bodies with internal variables to cyclic loading is deduced from the features of the deformation path at the post adaptation stage: shakedown is possible for taken values of internal variables, if there exists a solution of a min-max problem. In some cases (perfectly plastic bodies, and isotropically strain hardening) the condition is also sufficient. In the last case the condition provides a method to determining the minimum value of hardening parameter which can be acquired under the given loading program. An example of application of the method is given.