Lorenz-84 system was proposed about four decades ago, however, there are almost no analytical results on the equilibria and their local stability. The first objective of this paper is to fill this gap. We discuss the possibility of the existence of multiple equilibria and establish the conditions for a given number of equilibria to exist by using algebraic methods of resultant. Furthermore, we derive the stability conditions on the parameters of the system by using symbolic methods for solving semi-algebraic systems. The second objective is to investigate the zero-Hopf bifurcation of the Lorenz-84 system. By using the averaging method, we provide sufficient conditions for the existence of one limit cycle bifurcating from a zero-Hopf equilibrium of Lorenz-84 system. Several examples and numerical simulations are presented to verify the established results.