condition ${\cal L}(\rho_\mathrm{c},T_\mathrm{c})=0$. The momentum flux is determined by $P_\mathrm{c}$ and $j$ from equations (\[const\]) and (\[const mom\]).

The properties of differential equation (\[integ eq\]) can be well understood as trajectories in the phase diagram $(\theta,d\theta/dx)$. From equation (\[integ eq\]), one can see that the spatially uniform state with thermally equilibrium (${\cal L}=0$) corresponds to a stationary point because $d\theta/dx=d^2\theta/dx^2=0$. The topological property of the CNM at $x=-\infty$ corresponds to saddle point [@Eetal92; @FS93]. Given $P_\mathrm{c}$ and $j$, we numerically integrate equation (\[integ eq\]) from $x=-\infty$ along one of the eigenvectors obtained from the Jacobi matrix of equation (\[integ eq\]) around the stationary point of the CNM.

Zel’dovich & Pikel’ner Solutions {#sec:ZP}
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First, we review the solutions connecting two thermal equilibrium states (CNM and WNM). We call the solutions derived by ZP69 the ZP solutions. The ZP solutions exist only in the pressure range of $P_\mathrm{min}<P_\mathrm{c}<P_\mathrm{max}$ (see figure \[fig:eq\]). Given $P_\mathrm{c}$, @SI01 determine $j=j_\mathrm{ZP}(P_\mathrm{c})$ as the eigenvalue problem so that the boundary condition $d\theta/dx=0$ is satisfied in the WNM at $x=\infty$.

For the case of a static solution, that is $j=v=0$, equation (\[integ eq\]) can be rewritten as $$\int_{\theta_\mathrm{c}}^{\theta_\mathrm{w}} \rho {\cal L}d\theta
        = \int_{T_\mathrm{c}}^{T_\mathrm{w}}\kappa\rho{\cal L}dT = 0
 \label{saturate}$$ (ZP69) 