Why are first integrals constant along characteristic curves? My lecturer in a PDE course seems to be using this fact. I found a nice explanation of what a first integral is here: Why are they called "first integrals"?. However, I am unable to connect this with the concept of a characteristic, or understand why first integrals are constant along characteristics. For context, this was used in explaining "alternative method of solution using characteristics", that is constructing a family of characteristic curves using two linearly independent first integrals of a pde.
Maybe this is relevant: What does constant along characteristic mean, but I don't see how.