There are quite a few cases of West Germans moving East to the German Democratic Republic (commonly called East Germany). Angela Merkel, the current Chancellor of the Federal Republic of Germany was born in Hamburg in 1954. Hamburg was in what was then West Germany and her father was a Lutheran pastor, shortly after her birth their family left West Germany so that her father could become the pastor of a church in Quitzow, Brandenburg and she was raised in the town of Templin — north of Berlin in East Germany. Political defectors include Bertoldt Brecht who was a Bavarian playwright who fled to America during World War II and then moved to East Berlin (then the Soviet occupied sector) after he had to testify in front of the House UnAmerican Activities Committee during the late 40s Red Scare in the United States, James Stuart who was a reporter for Reuters and a veteran of the Spanish Civil War who defected because of his opposition to West German rearmament in 1950 and spent his life producing English language pro-East German propaganda and playing the part of an Englishman in numerous East German films.

From an earlier answer of mine

This is far from comprehensive, but gives a general idea that some high-profile defections did occur. Most of the defections by Germans from the FRG to the GDR were connected to Stasi intelligence. For example, Hans Joachim-Tiedge was one of the famous cases from the 1980s and his rationale for defecting is still nebulous.

But there were other cases of defection that had little to do with espionage. Manfred von Brauchitsch defected to the GDR to escape his tax debts and the GDR offered him financial incentives to do so. Even more enigmatic is the case of Otto John who defected to the GDR in 1954 partly out of the conviction that West German rearmament and the continued presence of Nazis within West German society ran counter to the ideal that German after 1945 had turned its back on National Socialism.

John's case highlights an important point about the Cold War in Germany, especially in its early decades. Although it is tempting to conceptualize the two Germanies as "democracy vs. totalitarianism," the distinction was less clear cut during the time. Although Adenauer's West Germany made public commitments to democracy, it possessed a high degree of social conservatism and one of the pillars of its political society was a fervent anticommunism. For many Germans that rejected Nazism, the latter was particularly troubling given that the FRG adapted a very lenient stance towards denazification. Adenauer cemented a silent compact that offered former Nazis (at least those who were not directly responsible for major crimes) a place in the new government provided that they kept their mouth shut about the previous regime and did not directly undermine democratic institutions. This was something that fed a great deal of ideological dissatisfaction among a number of West Germans. Although things like the 1953 East Berlin uprising and the Berlin Wall soiled the image of the GDR and communism, a reaction against the social and political strictures of the Adenauer state was a consistent theme within the FRG throughout the Cold War.

There were other examples of migration on the basis of ideological affinities or a rejection of American cultural hegemony. Wolf Biermann left Hamburg for the GDR because he felt that the GDR stood for, as he put it in a recent interview, "the better Germany." Nor should this cultural or ideological appeal of a socialist Germany be entirely discounted; Victor Klemperer, whose diaries are an especially perceptive window into the Third Reich, elected to stay in the GDR on account of his dislike of the FRG's Christian conservatism and the crass materialism of its American patron. Klemperer's diaries from the postwar period show little love for the SED's dictatorship, but he saw it as the "lesser evil."

There were other, much less high-profile West-East migrations. One facet of German division was that it did divide families and some emigration did occur to take care of an elderly parent or other relation. Such family-based emigration was typically East to West, especially since the GDR government did not want to pay for pensioners in an already burdened planned economy. Other West-East migrants were former GDR citizens who found that life in the West was less romantic than the image of the Wirtschaftswunder presented. There was also a jump in West-East migration when the FRG introduced conscription for the newly formed Bundeswehr. Although Bonn's conscription did have provisions for conscientious objectors, these provisions were not as lax as they would be in later years. By the same token, one of the push factors for West-East flight in the mid-1950s was also fear that the NVA would enact its own form of conscription, showing that the war-inspired ohne mich (without me) sentiment transected the Iron Curtain in Germany. But West to East migration levels were always much lower even when there were similar factors at play. In the case of conscription, West Berlin whose Free University and exemption from conscription acted as a sump for ohne mich types. But such emigration to the GDR did occur, even if at a much lower rate than Republikflucht in the other direction.

Sources

Frei, Norbert. Adenauer's Germany and the Nazi Past: The Politics of Amnesty and Integration. New York: Columbia University Press, 2002.

Kundnani, Hans. Utopia or Auschwitz?: Germany's 1968 Generation and the Holocaust. New York: Columbia University Press, 2009.

Major, Patrick. Behind the Berlin Wall: East Germany and the Frontiers of Power. Oxford [u.a.]: Oxford University Press, 2011.