  In this paper, we consider parametric ideals and introduce a notion of
comprehensive involutive system. This notion plays the same role in theory of
involutive bases as the notion of comprehensive Groebner system in theory of
Groebner bases. Given a parametric ideal, the space of parameters is decomposed
into a finite set of cells. Each cell yields the corresponding involutive basis
of the ideal for the values of parameters in that cell. Using the Gerdt-Blinkov
algorithm for computing involutive bases and also the Montes algorithm for
computing comprehensive Groebner systems, we present an algorithm for
construction of comprehensive involutive systems. The proposed algorithm has
been implemented in Maple, and we provide an illustrative example showing the
step-by-step construction of comprehensive involutive system by our algorithm.
