Short of refinishing, nothing is going to make it look new but, Wood Elixir could go a long way toward making it look acceptable. If the furniture is 60 years old and has the factory finish, then it is lacquer. The only thing that could cause this type of damage in lacquer is a combination of high heat and humidity over a long time. The other possibility is that this is not the original finish but it is a quick drying finish (shellac?) applied over the lacquer without properly preparing the surface. To solve this, apply Wood Elixir with steel wool. Wipe dry and buff.