TABLE OF CONTENTS Exercise 6.1
Adding side by side the three equations, we get 
. Hence every solution of the system satisfies x 1+ x 2+ x 3= c 1. So, one prime integral is the function 
, defined by U 1( x 1, x 2, x 3)= x 1+ x 2+ x 3. Multiplying the equation of rank i with x i , i=1, 2, 3, and adding the equalities thus obtained, we deduce 
. Hence the function 
defined by 
is also a prime integral. Since
it follows that U 1, U 2 are independent about any non-stationary point. Indeed, let us observe that ( x 1, x 2, x 3) is non-stationary if and only if x 1 ? x 2, or x 1 ? x 3, or x 2 ? x 3, situations in which the rank of the matrix above is 2.
From the system, we deduce 
and 
. So, the functions 
, i=1, 2, defined by U 1( x 1, x 2, x 3)= x 1 ? x 2+ x 3, and 
respectively, are prime integrals for the system. The only stationary points of the system are of the form (0, 0, x 3). Since the rank of the matrix
is 2 at every point ( x 1, x 2, x
