To sum up, in one sentence, the answer is no; not all brand-name medicines have a generic counterpart. Medicine companies can introduce generic versions only after they've been thoroughly tested by the manufacturer and approved by the U.S. Food and Drug Administration.
When is a medicine turned into generic medicine?
A generic alternative cannot be launched if patents protect a medicine. Sometimes a generic version is approved and available in a specific country or countries while unavailable in others.
However, with the growing emphasis on generics, multiple generic versions of a brand name medicine are often developed around about the time it becomes "off patent."
However, sometimes, even after a patent has expired, there is no generic alternative.
You can find generic medical store franchises these days and search for the required medicine right in the store.