Two trees, never the same equation. An Evidence Press briefing. Lotka–Volterra equations describe how interacting quantities grow and shrink: predators and prey, competing species, chemical concentrations. Some of them are also studied as integrable systems, where hidden conserved quantities make the motion unusually orderly. Each variable in such an equation has an invariant plane: a population that starts at zero stays at zero. Some equations have extra invariant planes, made of combinations of the variables. These are called linear Darboux polynomials, and they matter because they produce conserved quantities, and because they decide which changes of coordinates keep the equation in Lotka–Volterra form. A few years ago, Quispel, Tapley, McLaren and van der Kamp built a family of equations from any tree, with one extra invariant plane for each edge of the tree. But if you change coordinates, the same equation can look like a different picture, a hypergraph rather than a tree. Van der Kamp conjectured that two different trees can never describe the same equation, for generic parameter values. This paper proves that conjecture, in a stronger form: one side needs to be generic, and the other can be anything. It also shows that the statement cannot be extended to every parameter value, because two tree families can share some special equations. The key to the proof is an exact list: a generic tree system on n variables has exactly two n minus one invariant planes, and the paper finds the precise condition on the parameters for this to hold. From that list it finds every coordinate system that keeps the equation's form, one for each spanning tree of a related graph, counts how many genuinely different pictures there are for paths and caterpillars, and gives a test for recognising a tree system in disguise. The planes also survive a popular numerical method, Kahan's discretisation. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on 22 October 2026, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.