CharacteristicQ(S, DepVars, 'options'='value')

Consider a problem in two independent and two dependent variables u(x, t), v(x, t), and the generic form of infinitesimals for this type of problem

By default CharacteristicQ returns, fast, an operator in its most abstract form, with a test-type for the value of  and not expanded; essentially, nothing is actually computed until you need it

This resulting characteristic is a function that can then be applied to an integer as large as the number of dependent variables of the problem, in this case two

You can instead request to CharacteristicQ for the sum in the mapping to be expanded before returning, or to avoid the check of type of the value of

Instead of passing the symmetry as a list of infinitesimals you can also pass the corresponding infinitesimal generator operator. You construct this operator with InfinitesimalGenerator

This is the same output as (4.4)

To request the output in function instead of jet notation use

Compare for instance this output with the output of

InfinitesimalGenerator, infinitesimals, PDEtools, ToJet