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I started Fringe at season 2 and I noticed a cow in Walter Bishop's Lab.

Considering the hassle of keeping a cow in a lab, notably the odour that should be awful, is there any purpose of keeping it?
Is there any back-story about it?
Does it even have a name?

From http://fringepedia.net/wiki/Gene

Gene is the name given to the cow requested by Walter Bishop during
the reopening of his lab at Harvard University. It is unknown who
named the cow Gene, but Olivia is the first to use the name.

Peter Bishop explains that since cows are genetically very similar to
humans, experiments on cows, simulate effects on humans; they are an
ethical test subject when using a human guinea-pig might be dangerous.

Walter asked for a cow in the pilot episode, "Pilot". It's unknown who named it Gene.

According to the transcript at Fringepedia:

ACT IV

Harvard Lab - Setting Up

OLIVIA: Dr. Bishop, I've ordered a standard forensics' work package. Is there anything else you need?

WALTER: Optical coherence tomograph for flesh study. Two thousand pounds of silicon. At least five anonymous blood samples from volunteer donors, micro-organism detector. NASA grade, of course. There's still NASA, yes?

OLIVIA: Yes.

WALTER: Oh, she's still here. This tank was the best. And a two-Year-old bos taurus.

ASTRID: A what?

PETER: A cow. He wants a cow.

WALTER: Purebred, not a crossbred, this is important. Mature weight 850 pounds, total fat average 2.37.

OLIVIA: Is he joking?

PETER: Genetically, humans and cows are separated by only a couple lines of DNA. So, what's an ethical test subject?

OLIVIA: Where'd you learn that? MIT?

PETER: No actually, I picked that up reading books. You should try it sometime. It's fun.

OLIVIA: (to Astrid) Get him the cow.

WALTER: Fantastic, thank you. Only thing better than a cow is a human. Unless you need milk. Then you really need a cow.

(Activity including Walter examining laboratory slides; Olivia reviewing photos of Flight victims; an unidentified, and unrelated, class in session; and the arrival of the cow.)

(...)

Harvard Lab - John Scott Arrives

(...)

CHARLIE: Is that a cow?

OLIVIA: Yeah, that's Gene.

Peter thinks Walter wants the cow for a test subject in the pilot, but Walter never actually says that himself. Walter actually mentions right then that you need a cow for milk, and we see Walter frequently drinking milk from the cow too. The only experiment I remember Walter ever doing with the cow involved trying to get the cow to produce chocolate milk!

So I think, if we read between the lines, Walter probably just wanted a cow for fresh milk and was pulling a fast one on the FBI :)