Abstract <p>The behavior of trajectories of solutions to piecewise linear second-order differential equations is studied. These equations are widely used in mechanics, electrical engineering, and automatic control theory. Of particular interest are the conditions for the emergence of limit cycles in the vicinity of the rest region of a piecewise linear second-order differential equation with a discontinuous switching line. It is established that if a region of rest (consisting of rest points) exists, then it remains inside the limit cycle. One of the primary tasks is to determine the region of rest that appears on the line of stitching solutions. In the course of the work, new relations are obtained that provide bounded solutions to piecewise linear equations. Using these new conditions, phase portraits are constructed that take into account the coefficients of the equations. Conditions under which there is no rest region are also found. To solve these problems, the method of stitching solutions from two half-planes is used.</p>