Geodesic
On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness
Serdica Mathematical Journal, Tome 34 (2008) no. 1, pp. 155-178.

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In this paper we prove that for non effectively hyperbolic operators with smooth double characteristics with the Hamilton map exhibiting a Jordan block of size 4 on the double characteristic manifold the Cauchy problem is well posed in the Gevrey 6 class if the strict Ivrii-Petkov-Hörmander condition is satisfied.
Keywords: Cauchy Problem, Non Effectively Hyperbolic, Gevrey Well-Posedness, Null Bicharacteristic, Hamilton Map, Elementary Decomposition, Positive Trace 
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Nishitani, Tatsuo. On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness. Serdica Mathematical Journal, Tome 34 (2008) no. 1, pp. 155-178. http://geodesic.mathdoc.fr/item/SMJ2_2008_34_1_a6/