The recently realized photonic-crystal Fano laser constitutes the first demonstration of passive pulse generation in nanolasers [Yi Yu, Nat. Photon. 11, 81 (2017)1749-488510.1038/nphoton.2016.248]. We show that the laser operation is confined to only two degrees of freedom after the initial transition stage. We show that the original five-dimensional dynamic model can be reduced to a one-dimensional model in a narrow region of the parameter space and it evolves into a two-dimensional (2D) model after the exceptional point, where the eigenvalues transition from being purely real to a complex conjugate pair. The 2D reduced model allows us to establish an effective band structure for the eigenvalue problem of the stability matrix and to explain the laser dynamics. The reduced model is used to associate an origin of instability with an unstable periodic orbit separating the stable steady state from the stable periodic orbit.