### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 096, 50 pages      arXiv:1703.01379      https://doi.org/10.3842/SIGMA.2017.096

### Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems

Hiroshi Kawakami
College of Science and Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa 252-5258, Japan

Received April 19, 2017, in final form December 07, 2017; Published online December 25, 2017

Abstract
This is the last part of a series of three papers entitled ''Four-dimensional Painlevé-type equations associated with ramified linear equations''. In this series of papers we aim to construct the complete degeneration scheme of four-dimensional Painlevé-type equations. In the present paper, we consider the degeneration of the Garnier system in two variables and the Fuji-Suzuki system.

Key words: isomonodromic deformation; Painlevé equations; degeneration; integrable systems.

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References

1. Babbitt D.G., Varadarajan V.S., Formal reduction theory of meromorphic differential equations: a group theoretic view, Pacific J. Math. 109 (1983), 1-80.
2. Boalch P., Simply-laced isomonodromy systems, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 1-68, arXiv:1107.0874.
3. Fuji K., Suzuki T., Drinfeld-Sokolov hierarchies of type $A$ and fourth order Painlevé systems, Funkcial. Ekvac. 53 (2010), 143-167, arXiv:0904.3434.
4. Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes, Acta Math. 33 (1910), 1-55.
5. Garnier R., Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes, Ann. Sci. École Norm. Sup. (3) 29 (1912), 1-126.
6. Harnad J., Dual isomonodromic deformations and moment maps to loop algebras, Comm. Math. Phys. 166 (1994), 337-365, hep-th/9301076.
7. Hiroe K., Oshima T., A classification of roots of symmetric Kac-Moody root systems and its application, in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat., Vol. 40, Springer, Heidelberg, 2013, 195-241.
8. Hukuhara M., Sur les points singuliers des équations différentielles linéaires, II, J. Fac. Sci. Hokkaido Univ. 5 (1937), 123-166.
9. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), 306-352.
10. Kapaev A.A., Lax pairs for Painlevé equations, in Isomonodromic Deformations and Applications in Physics (Montréal, QC, 2000), CRM Proc. Lecture Notes, Vol. 31, Amer. Math. Soc., Providence, RI, 2002, 37-48.
11. Kapaev A.A., Hubert E., A note on the Lax pairs for Painlevé equations, J. Phys. A: Math. Gen. 32 (1999), 8145-8156.
12. Kawakami H., Matrix Painlevé systems, J. Math. Phys. 56 (2015), 033503, 27 pages.
13. Kawakami H., Four-dimensional Painlevé-type equations associated with ramified linear equations I: Matrix Painlevé systems, arXiv:1608.03927.
14. Kawakami H., Four-dimensional Painlevé-type equations associated with ramified linear equations II: Sasano systems, arXiv:1609.05263.
15. Kawakami H., Nakamura A., Sakai H., Degeneration scheme of 4-dimensional Painlevé-type equations, arXiv:1209.3836.
16. Kawamuko H., On the Garnier system of half-integer type in two variables, Funkcial. Ekvac. 52 (2009), 181-201.
17. Kimura H., The degeneration of the two-dimensional Garnier system and the polynomial Hamiltonian structure, Ann. Mat. Pura Appl. 155 (1989), 25-74.
18. Levelt A.H.M., Jordan decomposition for a class of singular differential operators, Ark. Mat. 13 (1975), 1-27.
19. Nakamura A., Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two, arXiv:1505.00885.
20. Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$, J. Math. Sci. Univ. Tokyo 13 (2006), 145-204.
21. Ohyama Y., Okumura S., A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations, J. Phys. A: Math. Gen. 39 (2006), 12129-12151.
22. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_{{\rm II}}$ and $P_{{\rm IV}}$, Math. Ann. 275 (1986), 221-255.
23. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{{\rm VI}}$, Ann. Mat. Pura Appl. 146 (1987), 337-381.
24. Okamoto K., Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\rm V}$, Japan. J. Math. (N.S.) 13 (1987), 47-76.
25. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation $P_{{\rm III}}$, Funkcial. Ekvac. 30 (1987), 305-332.
26. Painlevé P., Sur les équations différentielles du second ordre à points critiques fixes, C. R. Acad. Sci. Paris 127 (1898), 945-948.
27. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
28. Sakai H., Isomonodromic deformation and 4-dimensional Painlevé type equations, Preprint, University of Tokyo, Mathematical Sciences, 2010.
29. Sasano Y., Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D^{(1)}_6$. II, in Algebraic Analysis and the Exact WKB Analysis for Systems of Differential Equations, RIMS Kôky^uroku Bessatsu, B5, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, 137-152, arXiv:0704.2367.
30. Suzuki T., A particular solution of a Painlevé system in terms of the hypergeometric function $_{n+1}F_n$, SIGMA 6 (2010), 078, 11 pages, arXiv:1004.0059.
31. Tsuda T., UC hierarchy and monodromy preserving deformation, J. Reine Angew. Math. 690 (2014), 1-34, arXiv:1007.3450.
32. Turrittin H.L., Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math. 93 (1955), 27-66.
33. Wasow W., Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. 14, Interscience Publishers John Wiley & Sons, Inc., New York - London - Sydney, 1965.
34. Yamakawa D., Middle convolution and Harnad duality, Math. Ann. 349 (2011), 215-262, arXiv:0911.3863.