Symmetries of differential-equations.4.

Abstract

By an application of the geometrical techniques of Lie, Cohen, and Dickson it is shown that a system of differential equations of the form [x^(r_i)]_i = F_i; (where r_i > 1 for every i = 1 , ... ,n) cannot admit an infinite number of pointlike symmetry vectors. When r_i = r for every i = 1, ... ,n, upper bounds have been computed for the maximum number of independent symmetry vectors that these systems can possess: The upper bounds are given by 2n _2 + nr + 2 (when r> 2), and by 2n_2 + 4n + 2 (when r = 2). The group of symmetries of ͞x_r = 0͞͞ (r> 1) has also been computed, and the result obtained shows that when n > 1 and r> 2 the number of independent symmetries of these equations does not attain the upper bound 2n_ 2 + nr + 2, which is a common bound for all systems of differential equations of the form x͞_r = F͞ (t, x͞, ... ,͞x (r - 1 ) when r> 2. On the other hand, when r = 2 the first upper bound obtained has been reduced to the value n_2 + 4n + 3; this number is equal to the number of independent symmetry vectors of the system ¨x͞ = 0͞, and is also a common bound for all systems of the form x͞ = F͞ (t, x͞, x͞).