Geodesic
On equations with subgroup constraints on solutions in free groups
Diskretnaya Matematika, Tome 7 (1995) no. 4, pp. 60-67

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It is proved that a series of problems consisting in determination for the given equation in a free group $F_n$ whether it has a solution with the first component belonging to some subgroup of $F_n$, are algorithmically unsolvable.The work is supported by the Russian Foundation for Fundamental Investigations, Grant 93–011–1552. 
@article{DM_1995_7_4_a4,
     author = {V. G. Durnev},
     title = {On equations with subgroup constraints on solutions in free groups},
     journal = {Diskretnaya Matematika},
     pages = {60--67},
     publisher = {mathdoc},
     volume = {7},
     number = {4},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1995_7_4_a4/}
}
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V. G. Durnev. On equations with subgroup constraints on solutions in free groups. Diskretnaya Matematika, Tome 7 (1995) no. 4, pp. 60-67. http://geodesic.mathdoc.fr/item/DM_1995_7_4_a4/