Geodesic
On spectra of computable models
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 4, pp. 93-97

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For every $n\in N$ we construct an example of theory with countably many countable models, such that its isomorphism type is defined by two dimensions. If we fix the second dimension, the first dimension gives the spectrum $\{1,\dots,n\}$. If we fix the first one then all models with this dimension are computable or are not computable simultaneously. 
@article{VNGU_2006_6_4_a5,
     author = {E. B. Fokina},
     title = {On spectra of computable models},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {93--97},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2006_6_4_a5/}
}
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E. B. Fokina. On spectra of computable models. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 4, pp. 93-97. http://geodesic.mathdoc.fr/item/VNGU_2006_6_4_a5/