For your first question,

How to find a gene without orthologs in closely related species

what gene exactly you want?

In this digital era you get hell lot of information, sit patiently and check you get what you need.

First decide your gene then go to the following link and search

http://questfororthologs.org/orthology_databases

http://cegg.unige.ch/orthodb6.

by seeing the links, find the similarity and avoid orthologs

"Orthologs are genes that have evolved from a common ancestor gene. These genes can be in different species or within the same species. Othologs can be identified by comparing either gene or amino acid sequences and looking for similarity. Learn more about building genetic trees at The History of Life (http://evolution.berkeley.edu/evosite/evo101/IIHistory.shtml). BLAST protein searches can scan all the proteins of known sequence and look for correspondence with your protein of interest".

Hope your part of question answered!!!

Yes, I know what orthlogs genes are.

That's the point: I do not know the gene/s. Initially I don't care about its function, either. It's about data mining. I want to find unique genes-sequences in a given genome, i.e., sequences with neither BLAST hits nor orthologous sequences in other related taxa. In bacteria LGT is not uncommon and hence this task is not straightforward. For instance sequences in humans which are absent in other primates are most likely also missing in other mammals. However, in bacteria that is not necessarily so because xenologous sequences are quite common

I