So it's as I thought.
The paper describes a process of identification a gene with novel sequence, and they were tring to "fish it" with primers of similar organisms.You can use the BLAST Align to see yourself.
The Pseudomonas sp. AMS8 LipAMS8 gene GenBank accession is HQ162821.1 (mentioned in the paper).
Neither of the primer pair is completely specific.
The first pair has a single mismatch in reverse primer, the second pair has only match from 7-18th nucleotide of the primer (not so much specific, but the 3' end is matched so it may amplify). The first pair only amplified a small portion of the gene and they wanted it sequenced.
Now the question is what do you want. Amplify the complete gene?
I found there is a primer sequence, that amplifies the whole sequence and is correct, its' the one down in the table for the cloning, the pTrcHis TOPO TA (TA cloning doesn't need any overhangs, so its' just simply primers for whole gene). You may use the accession to check that it's really specific.
But also if you are amplifying not from a single colony, you may need to check for specifity of the primers in the genetic background you have.