Invariant algebraic surfaces of the Rabinovich system

Global analytic integrability of the Rabinovich system

“…By using some algebraic methods in 1991, Giacomini et al [2] obtained more additional four integrals again for particular values of the parameters. In 2000 Zhang [10] and in 2003 Xie and Zhang [9] studied the Darboux integrability of the Rabinovich systems. Here we shall study its analytical integrability.…”

“…HenceḠ is a global analytic first integral of system (9) with v = 0. Again by Lemma 5 we get thatḠ =Ḡ(G 1 , G 2 ).…”

“…If f = f (x, y, z, v) is a global analytic first integral of system (9), then f only depends on the variable v.…”

“…Since f is a global analytic first integral of system (13), f (x, y, z, 0, v) is a global analytic first integral of system (9). So we are in the assumptions of Lemma 9, applying it we get that f (x, y, z, 0, v) = T (v), that is, T is a global analytic function of v. Therefore using the convergent power expansion series of f (x, y, z, h, v) in a neighborhood of the origin (0, 0, 0, 0, 0) we can write f (x, y, z, h, v) = T (v) + hg, where g = g(x, y, z, h, v) is a convergent power series in a neighborhood of (0, 0, 0, 0, 0).…”

“…where in the last expression we have used that T is a first integral of system (9) with v = 0, and it is evaluated along a solution of system (1) with v 1 = v 2 = v 3 = 0. After dividing this last equality by v 3 , taking v 3 = 0 and definingḠ = G(x, y, z, 0) we have that…”