  The main objective of this paper is the following two results. (1) There
exists a computable bi-orderable group that does not have a computable
bi-ordering; (2) There exists a bi-orderable, two-generated recursively
presented solvable group with undecidable word problem. Both of the groups can
be found among two-generated solvable groups of derived length $3$.
  (1) answers a question posed by Downey and Kurtz; (2) answers a question
posed by Bludov and Glass in Kourovka Notebook.
  One of the technical tools used to obtain the main results is a computational
extension of an embedding theorem of B. Neumann that was studied by the author
earlier. In this paper we also compliment that result and derive new
corollaries that might be of independent interest.
