A question about conservation of momentum (Lagrangian)

The image seems to be not showing here is the link : http://imgur.com/QhYG54l


What does Landau mean here by the Lagrangian remaining ''unchanged''. Is it the value of the lagrangian as a function that may not change or is it the form that may not change?

Also how does this curly delta differ from the normal differential symbol?

I'm confused because what follows seems like a derivation as if the value of the Lagrangian as a function can't change, but then why all those curly delta's from variational calculus?

Thanks