Low scattered linear orders

In 1998 R. Downey formulated a problem: to describe a property P of classical order types, which guarantees that if L is a low linear order and P holds for the order type of L then L is isomorphic to a computable linear order. We find a new such property P. Also, we give an upper bound on a complexity of an isomorphism between computable and low copies and show that this bound is sharp.