In this paper, we prove that the three-dimensional \(\Lambda \) CDM model of the universe depending on the parameters \(a,b \in [1,4]\) is completely integrable, providing two independent first integrals. Moreover, when \(a=b\) we describe the dynamics of the mathematical \(\Lambda \) CDM model in the positive octant of \(\mathbb {R}^3\) . The dynamics of the mathematical \(\Lambda \) CDM model when the parameters a and b are equal has no physical meaning because the parameter a can not be equal to b, but the knowledge of this dynamics provides some information on the dynamics of the physical \(\Lambda \) CDM model when the parameters a and b are close. Moreover, the dynamics of the physical \(\Lambda \) CDM model only has meaning in the region of the positive octant having all their coordinates greater than or equal to one.
