Wandering of solutions of two-dimensional diagonal and triangular systems of differential equations

We consider some classes of two-dimensional diagonal and triangular linear nonautonomous systems of differential equations with bounded coefficients. It is shown that the upper, as well as the lower, walk exponents and wandering exponents of all their nontrivial solutions are equal to zero, except, possibly, the upper wandering exponent for a triangular system (an example is constructed in which the latter exponent is positive).