Geodesic
Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems
Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 848-857

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We generalize the notions of Laplace transformations and Laplace invariants for systems of hyperbolic equations and study conditions for their existence. We prove that a hyperbolic system admits the Laplace transformation if and only if there exists a matrix of rank $k$ mapping any vector whose components are functions of one of the independent variables into a solution of this system, where $k$ is the defect of the corresponding Laplace invariant. We show that a chain of Laplace invariants exists only if the hyperbolic system has a entire collection of integrals and the dual system has a entire collection of solutions depending on arbitrary functions. An example is given showing that these conditions are not sufficient for the existence of a Laplace transformation. 
@article{MZM_2003_74_6_a4,
     author = {A. V. Zhiber and S. Ya. Startsev},
     title = {Integrals, {Solutions,} and {Existence} {Problems} for {Laplace} {Transformations} of {Linear} {Hyperbolic} {Systems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {848--857},
     publisher = {mathdoc},
     volume = {74},
     number = {6},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a4/}
}
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A. V. Zhiber; S. Ya. Startsev. Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems. Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 848-857. http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a4/