The 16th Hilbert Problem for Discontinuous Piecewise Linear Hamiltonian Saddles and Isochronous Centers Separated by a Straight Line

We provide the maximum number of limit cycles for discontinuous piecewise differential systems separated by a straight line and formed by a linear Hamiltonian saddle and one of the four families of quadratic isochronous centers. Hence we have solved the extension of the 16th Hilbert problem to such piecewise differential systems and for two of these four classes of discontinuous piecewise differential systems the obtained upper bound for the maximum number of limit cycles is reached.