On the number of isolating integrals in systems with three degrees of freedom

Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: $$left{ begin{gathered} x_1 = x_0 + a_1 {text{ sin (}}x_0 {text{ + }}y_0 {text{)}} + b{text{ sin (}}x_0 {text{ + }}y_0 {text{ + }}z_{text{0}} {text{ + }}t_{text{0}} {text{)}} hfill y_1 = x_0 {text{ + }}y_0 hfill z_1 = z_0 + a_2 {text{ sin (}}z_0 {text{ + }}t_0 {text{)}} + b{text{ sin (}}x_0 {text{ + }}y_0 {text{ + }}z_{text{0}} {text{ + }}t_{text{0}} {text{) (mod 2}}pi {text{)}} hfill t_1 = z_0 {text{ + }}t_0 hfill end{gathered} right.$$ We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).