Abstract: In this paper some new results concerning analysis and synthesis of time optimalcontrol systems are given. We assume that the motion of controlled object is describedby linear ordinary differential equations with variable coefficient, and the final states ofthe system form a convex region of phase space. Geometric properties of the iso-chronous region of these systems in phase space and the facts connected with these whichhavc fundamental meaning for the solution of the synthesis problem are discussed. More-over, the facts which are formulated in theorems have also their independent interest.At the end of this paper the method of synthesis of the optimal control system with con-stant and variable coefficients and two illustrative examples are given.