The recent trademark for the name "Foxyshot" has been found to be for a mobile phone game released in 2005. It was available for download via Sega's "Sonic Café" service.
Here is a translation of the plot:
"A terrorist group hijacked the city! Calamity Jane must wipe them out in a magnificent gun fight and save the hostages!"The gameplay is quite simple. A terrorist appears in one of nine squares - to shoot them you must press the corresponding number on your keypad. You can hide behind a wall Time Crisis-style, and must reload when you have run out of ammo.
Hostages sometimes appear in the windows, so try not to shoot them! At the end of a level there is a boss - defeat him to proceed. There are three levels in total, each in a themed area: a plant, a nightclub and a movie theatre.
"Future Police Patochaser" still appears to be an upcoming game, but we'll just have to wait and see.
[Source: Sega Nerds]