New York state is currently without a Lieutenant Governor. The Governor elected in 2006, Eliot Spitzer, resigned on March 17, 2008, so the Lieutenant Governor, David Paterson, became Governor, and is Governor today. On July 8, 2009, Governor Paterson appointed Richard Ravitch as the new Lieutenant Governor. However, the State Supreme Court, Appellate Division, ruled on August 20 that the Constitution does not permit such an appointment. Here is the 10-page decision. Thanks to ElectionLawBlog for this news. The case is Skelos v Paterson, 2009-06673, Brooklyn Appellate Division. There may be an appeal to the State Court of Appeals.
The only real world situation would be to elect both on a ticket and then allow the elevated Lt. Governor/ Governor to select his replace ment. [Hopefully the new Governor will not put self, party, and corruption ahead of the public good like new [and bad] President Gerald ‘Pardon Me Boy’ Ford in the 1970s!]
Why does New York need a Lieutenant Governor? Surely it has a provision for a succession if something should happen to Patterson; and the Senate doesn’t need a presiding officer.
In Texas, the Senate appoints a new Lieutenant Governor from among its members, in case of vacancy. This was used in 2001, when Governor Bush resigned to become President.