A Banach-scales approach is used for studying analytic solutions of systems of linear partial differential equations satisfying the Leray-Volevich conditions. Results on the Cauchy problem lead to a continuation result which is of the same type as the well known result of Widder  on analytic solutions of the heat equation.
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W. Watzlawek, Remarks on Analytic Solutions of Leray-Volevich-Coupled Systems. Z. Anal. Anwend. 16 (1997), no. 2, pp. 363–375DOI 10.4171/ZAA/767