Let \(L\) be a Picard-Vessiot extension of \(F\), \(R\) be a ring of Picard-Vessiot elements of \(L\) over \(F\) and \(G= \text{Gal} (L/F)\). Suppose that \(R= F[y_1,\dots,y_n]= F[y]\), \(G(V)\subset V\), where \(V\) is a linear envelope of \(y\) over the constants of \(F\), and let \(I\) be a defining ideal of \(y\) in \(F[y]\). The author presents in the paper an effective algorithm for seeking the generators of \(I\). For the partial case \(G= \text{PSL}_2\), \(F= C(x)\), these generators are obtained in an explicit form. Also described are the connections of the discussed problem with the problems of algebraic independence of the values of \(E\)-functions [see \textit{A. B. Shidlovskii}, Transcendental numbers, De Gruyter Stud. Math. 12 (1989); translation from the original, Nauka, Moscow (1987; Zbl 0629.10026)] and looking for the first integrals of linear differential equations [\textit{J. A. Weil}, Lect. Notes Comput. Sci. 948, 469-484 (1995; Zbl 0885.12005)].