Painting - repainting galvanized surfaces ?

I have acquired some older wrought iron style lawn furniture. It is peeling badly and under the chips it appears to be a galvanized finish. How can I remove the old finish and repaint and avoid the chipping that is is now doing? It is not rusting which also makes me think it is galvanized.
splinters

I was going to recommend sandblasting, until I noticed that you said that the pieces might be galvanized. If that is the case, stripping would be better so as not to remove the galvanized finish.

It strikes me that the finish chips off of metal furniture because it is on such a rigid substrate. Any blow to the piece would cause the finish to chip because it will flex more than the metal substrate. I watch my furniture chip slowly as it is bumped into other pieces or the grandchildren knock it over. I refinished it all last year.

In the paint lines I have familiarity with (Rustoleum and Ace) there are specific primers within the line for galvanized metal.These primers are also used for aluminum and are specifically labeled as such.After application any oil base enamel would likely work.