Exact solutions of the simplified Keller–Segel model

•Maximal algebra of invariance of a simplified Keller-Segel system is found.•The algorithms for constructing traveling wave and self-similar solutions of this system are presented.•Exact solutions in explicit forms are found and their properties examined.•Applicability of the solutions obtained for chemotaxis modeling is discussed.

Lie symmetries of a simplified Keller–Segel system are found and applied for construction of exact solutions. The algorithms for constructing all possible traveling wave and self-similar solutions of the system in question are presented. Several families of such solutions in an explicit form are found, their properties examined and possible applicability for chemotaxis modeling is discussed.