Definately not an oxypetalum - wrong shape "branches" - but what concerns me is the rust coloured spots on them. Does your whole plant look like this? Do you keep this plant outside during the summer? It could be the result of overwatering or it could be an infection of some sort. If you can take any cuttings of unblemished branches do so now, and leave them about 2 weeks before trying to get them to root. I find they root quite quickly in pure coir. You might want to try repotting the mother plant in different soil as suggested by the previous poster. Remove as much of the current soil as you can from the roots before repotting, and remove all the damaged foliage - it will not recover. Just in case, do not put them in your compost or anywhere near other plants, and keep the repotted plant isolated. Give it bright light, but not direct sun, and do not water for a couple of weeks. You might also want to check out http://www.epiforums.com
I suspect you are referring to this post: http://www.botanicalgarden.ubc.ca/forums/showthread.php?t=31168&highlight=oxypetalum The photo I've attached was botanically confirmed as Epiphyllum oxypetalum and has very similar blades. In nature these species often take on varying forms and it is quite difficult to declare a species from the blade form alone. The plants within my tropical atrium are often used as teaching tools by professors at nearby universities. Scientists typically prefer to use the flower as a form of positive identification. Your advice on removing the damaged blades is very good since they will never heal. And I certainly believe the plant needs to be repotted. My guess is the damage is from excess water and root damage.