0708-1300/Barnie the polar bear

Barnie is a polar bear. Every morning, Barnie walks to the south x steps then x steps to the east, x to the north and finally x to the west. He do this to keep its shape. He is a little bit lacy and every day he walks for a smaller time, so x is decreasing every time. Of course, every day Barnie ends his trip at home.

But, we are going to change a little bit Barnie's fairy world. We are going to give Barnie two linearly independent vector fields ($X$ and $Y$) on the earth and Barnie should follow the local flows generated by these vector fields during his journey.

Pour Barney! It can happen that at the end of his walk he wont end at home and he will have to walk a little more to go home. But not too much. In fact, all the ends of his walks form a curve which is almost constant at home!! (its derivative is zero). The only problem is the second derivative which is twice $[X,Y]$ at home. Even more if $[X,Y]$ is zero then Barnie will be happy again, ending every day at home at the end of his walk.