Suppliers like Digikey have a category for slider potentiometers. There are only 155 items in that one. If you know the resistance of your broken part (you can measure it, also check if it's linear or logarithmic), it will narrow the search a lot. When you arrive at the short list, measure dimensions of the pin pattern of the old pot. Look through the datasheets for a compatible pattern.
If you don't get lucky, you can do surgery on the board. Your PCB by itself is very primitive: single sided, throughole components. In principle, you could drill new holes for a brand new slider pot and fly wires to it.
By the way, what kind of instrument are you repairing?