The density-density plot of the critical lines of the van der Waats equation at the van Laar point is analyzed through its algebraic properties. It is shown that this curve is an irreducible expression of the fifth degree of genus one. In addition, we show the topology of the second branch, i.e., the T= 0 solution, which will interact with the first branch as soon as the energy parameters are slightly different from the van Laar values. Finally, we analyze the behavior of the van der Waals equation near the point at which liquid-liquid separation takes place.