Geodesic
The existence of a~fundamental solution of power growth for systems of difference equations
Matematičeskie zametki, Tome 18 (1975) no. 5, pp. 731-734.

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It is proved that any nondegenerate system of difference equations of finite order with constant coefficients has a fundamental solution, increasing not faster then some power of the distance from the origin. A proof is obtained by the reduction to a known similar result for the case of one equation. 
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     title = {The existence of a~fundamental solution of power growth for systems of difference equations},
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A. Ya. Belyankov. The existence of a~fundamental solution of power growth for systems of difference equations. Matematičeskie zametki, Tome 18 (1975) no. 5, pp. 731-734. http://geodesic.mathdoc.fr/item/MZM_1975_18_5_a9/