Geodesic
Turing Machines Connected to the Undecidability of the Halting Problem
Matematičeskie zametki, Tome 71 (2002) no. 5, pp. 732-741.

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The problem of finding a Turing machine with undecidable halting problem whose program contains the smallest number of instructions is well known. Obviously, such a machine must satisfy the following condition: by deleting even a single instruction from its program, we get a machine with decidable halting problem. In this paper, Turing machines with undecidable halting problem satisfying this condition are called connected. We obtain a number of general properties of such machines and deduce their simplest corollaries concerning the minimal machine with undecidable halting problem. 
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L. M. Pavlotskaya. Turing Machines Connected to the Undecidability of the Halting Problem. Matematičeskie zametki, Tome 71 (2002) no. 5, pp. 732-741. http://geodesic.mathdoc.fr/item/MZM_2002_71_5_a8/

[1] Maltsev A. I., Algoritmy i rekursivnye funktsii, Nauka, M., 1965

[2] Pavlotskaya L. M., “Razreshimost problemy ostanovki dlya nekotorykh klassov mashin Tyuringa”, Matem. zametki, 13:6 (1973), 899–909 | MR | Zbl