Earth has the van Allen radiation belts. Which other planets have one? Can it be determined if distant planets in other solar systems have one?
Mars does not have a magnetic field of any strength and so does not have a van Allen belt (and this is a serious problem for potential human exploration of the planet). Similarly, neither Venus nor Mercury (nor the Moon, which is arguably part of a two-planet system with the Earth) have a van Allen belt.
The gaseous giants do, however, have radiation belts which are analogous to Earth's van Allen belt.
Spotting aurorae on distant exoplanets would be an indication of a strong magnetic field and thus of the likelihood of a radiation belt. I am not aware that such aurorae have been detected but distant aurorae have been seen around a brown dwarf - a substellar object somewhat bigger than a planet (see http://www.theguardian.com/science/2015/jul/29/astronomers-find-aurora-a-million-times-brighter-than-the-northern-lights).
Alternatively, the radio emissions from high energy electrons trapped in planetary radiation belts might be detectable - this was how Jupiter's radiation belt was discovered around 60 years ago. Work seems to be underway to detect such radiation: https://skaoffice.atlassian.net/wiki/download/attachments/22183971/PoS-exopla-AASKA14.pdf?version=1&modificationDate=1403003959989&api=v2
Yes, according to this recent news an aurora has been found on the brown dwarf LSR J1835+3259 which is 18½ light years from here. I suppose that an aurora which is a million times more powerful than those on Earth comes together with some kind of van Allen Belts, but real astronomers maybe have more imagination. An M class brown dwarf is maybe not the typical "planet", but it's more like Jupiter than the Sun. (All planets certainly don't have it, the headline is confusing).