On differential substitutions for evolution systems

For the most known differential substitutions relating scalar evolution equations, the sets of the equations admitting them consist of not finitely many equations but they form families parametrized by an arbitrary function. Some differential substitutions for evolution systems also have a similar property. In the present paper we obtain necessary and sufficient conditions for a differential substitution to be admitted by a family of evolution systems depending on an arbitrary function. We also give explicit formulae for finding the corresponding family of evolution systems in the case when these conditions are satisfied.
As an example, the family of systems admitting a multi-component Cole–Hopf substitution is constructed. We demonstrate that this family contains all linear systems, whose right hand sides contain no terms independent of the derivatives. As a result, we obtain a set of C-integrable systems of arbitrary high order. Another example considered in the paper is a multi-component analogue of the substitution $v=u_x+\exp(u)$. We show that this multi-component substitution is also admitted by a family of evolution systems depending on an arbitrary function.