Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of coupled ordinary differential equations of the form

$$
\frac{dx_i}{dt} = \displaystyle \sum_{j,k,l \in I} K^{kl}_{ij} (x_k x_l - x_i x_j)
$$

where each variable $x_i$ represents the total amount of particles with velocity $v_i$ for all $i \in I$, and $K^{kl}_{ij}$ are nonnegative constants that are nonzero only if the conservation relations 

$$
v_i+v_j = v_k+v_l, \ \ \ \ \ \  |v_i|^2+|v_j|^2 = |v_k|^2+|v_l|^2
$$

are satisfied. 

(In other words, make the standard assumption that collisions between particles obey conservation of momentum and conservation of energy.)

My question is: was it proved already that 

**(i)** *such a system of ODEs has a specific set of positive equilibrium points (i.e., has the set of equilibrium points been fully characterized)*, and that 

**(ii)** *all positive solutions converge to these equilibria?*

By the way, I am not interested in "weird" cases, where there are too few collisions (or no collisions) because the set $\{v_i |\ i \in I\}$ is too small etc. Instead, I am most interested in "reasonable" cases, e.g., where the set $\{v_i |\ i \in I\}$ consists of all velocities of the form $\mathbb Z^n \cap B(0,V_{max})$, i.e., all velocities in a nice regular grid that are contained in a ball of radius $V_{max}$, and under the assumption that $V_{max}$ is large enough to allow for "a lot" of collisions to occur; for example, take $n=2$ or $n=3$, and take $V_{max} = 10$, or $V_{max} = 100$. 

My guess was that this question has been answered long time ago (for all "reasonable" cases), but I looked pretty hard for references and I could not find it. By the way, I am aware that it is very easy to prove an H-theorem in this setting; but this does not (in my opinion) immediately answer my question, because *(i)* some nontrivial Diophantine equations need to be solved to find the set of equilibria, and *(ii)* the corresponding Lyapunov function is *not* infinite on the boundary.