What They Saw
They grew A. vinelandii OP (aka CA) and did genetic transformations with it, knocking out the nifDK genes (which encode the Mo dinitrogenase) by inserting a kanamycin resistance cassette by homologous recombination.
They plated transformant colonies on plate with or without kanamycin and with or without fixed nitrogen (so four different kinds of plate). The wild-type of course could grow on either plate without kanamycin but neither plate with it. They also saw two kinds of mutant phenotype: one that could grow with fixed nitrogen either with or without kanamycin, and another that could grow on all plates, with or without antibiotic or fixed nitrogen.
They tried to confirm this using PCR across the insertion, and restriction digestions followed by electrophoresis. The primers they report seem to be appropriate for getting the Mo nitrogenase genes.
Trying to figure out what they did, this paper seems to have lots of problems with reporting exactly what kind of digestions/cloning they did... sites they claim to have used don't seem to exist in my copy of the genome, or aren't in the right place, or are in too many places. So who knows what's actually going on genetically with this strain they isolated.
The gel they show from the PCR shows two bands in the kan-resistant, N-fixing mutant, and the bands seem to be the right size to correspond to sequences with and without the resistance cassette insertion. They claim to have sequenced the region but don't report the sequence in the paper.
What This Means
The authors conclude that A. vinelandii is exhibiting behavior suggesting multiple copies of a single gene locus (both with kan resistance and with nitrogen fixation). While it does seem to exhibit both phenotypes (assuming no contamination with multiple strains), I'm not sure the interpretation is clear. A. vinelandii has multiple different enzymes capable of fixing nitrogen, so a mutation in their regulation could explain that phenotype, though it wouldn't explain the bands on the gel. Still, it would be good to see this study replicated; unfortunately, the methods are explained poorly.