On Tuesday, the European Union's highest court stepped in and ruled in Slovakia's favor over the decision to bar Solyom. But some other issues -- dating back centuries -- are less easy to resolve. Here's a look at some of the main points of contention.
SLOVAKIA - OR UPPER HUNGARY?
For centuries, Hungary ruled large swathes of central Europe until losing two-thirds of its territory and half its population in the peace treaties at the end of World War I. In fact, the modern-day Slovak capital, Bratislava, was the capital of the Kingdom of Hungary for nearly 250 years until 1783. Some Hungarians still refer to southern Slovakia as "Upper Hungary" or the "Highland." That's enough to infuriate their neighbors to the north who proudly split from then-Czechoslovakia in 1993 and are now a relatively prosperous nation and member of the eurozone.
Some 460,000 ethnic Hungarians live in Slovakia -- and Solyom, the Hungarian president in 2009, was planning to visit them to attend a ceremony to unveil a statue to St. Stephen, Hungary's first king. Slovakia deemed such a visit provocative and told him he wasn't welcome. The dispute set a precedent. In its ruling Tuesday, the court said presidents don't enjoy the same rights as other EU citizens who can travel freely in Europe's borderless area. Many Hungarians and Slovaks say much of the tension between their countries is artificially fueled by politicians looking to attract voters. In 2010, a few days after leaving office, Solyom kept a promise made on occasion of his ban and went to Slovakia to lay a wreath at the foot of the St. Stephen statue unveiled without him a year earlier.
In 2009, Slovakia passed a language law which, while aiming to protect the national language, also curtailed the use of minority languages, including Hungarian. Billboards had to be in Slovak and some ethnic Hungarians were harassed for speaking in their mother tongue in public. Offenders faced fines of as much as 5,000 euros ($6,550) -- although the law, since considerably softened, was rarely enforced and Slovakia said it applied only to businesses and official public matters. The ban stoked tensions, with Hungary's top leaders lodging complaints and the far-right Jobbik party setting up a partial roadblock on the border with Slovakia. A protest at a soccer stadium in Slovakia attracted 10,000 people.
Slovak Prime Minister Robert Fico and his Hungarian counterpart Viktor Orban met last month in an effort to mend ties. They opened a community center for ethnic Slovaks living in the Hungarian town of Pilisszentkereszt (Mlynky in Slovak) and struck a deal to build a new bridge to connect the border towns of Komarno (Slovakia) and Komarom (Hungary), the same place where Solyom was turned back. It triggered some angry responses in online chat forums, though, with one poster saying it would be unwise to trust such gestures; another said the bridge was an ideal route for tanks.
Fico's recent return to power led some to fear that tensions between the two countries would escalate again, as he and Orban -- both populists -- would amplify their rhetoric and exploit their differences. So far, however, it has been rather smooth sailing and the meeting last month near Budapest for Slovaks in Hungary went off without a hitch.
BORDERLESS EUROPEAN UNION
One of the reasons many Hungarians supported their country's entry into the European Union was the lure of travel without border controls. They thought that would give Hungarians in neighboring countries like Slovakia and Romania easier access to the motherland. Since 2011, Hungary has allowed Hungarians living in neighboring countries to obtain Hungarian citizenship and passports with relative ease. Slovakia has objected strongly to the move and passed a law in 2010 prohibiting Slovaks from holding double citizenship. As a counter measure, Slovakia stripped Slovak citizenship from a handful of members of its Hungarian minority, as well as members of other minorities.
WHO'S FARING BETTER?
While Hungary jumped ahead of the rest of the region during the first years after communism, attracting much foreign investment and boosting its economy, Slovakia celebrated its own boom in the past decade and was labeled the "Slovak tiger." Hungarians took notice and hoped to become the "Pannon puma," an alliterative reference to Pannonia, a province of the Roman Empire which included parts of today's Hungary. Instead, Slovakia is already part of the eurozone, while in 2008 Hungary became the first EU country to receive a bailout from the International Monetary Fund and its economy is struggling with recession.