Two distinct degenerate Hopf bifurcation phenomena associated with autonomous lumped-parameter systems are explored in great detail via the intrinsic harmonic balancing method. It is assumed that the Hopf’s transversality condition is violated and certain other conditions prevail. In one of the cases, the system exhibits a cusp shape bifurcation path which exists for either positive or negative values of the system parameter. On the other hand, the second case is concerned with a tangential bifurcation phenomenon which may not be exhibited unless an additional condition is satisfied. This existence condition is obtained in the course of analysis. The distinctive feature of the paper is that the results concerning the bifurcating paths and limit cycles are given in general, explicit forms which are expected to be very useful in a variety of applications.