Preface -- I Plane Power Geometry -- 1 Plane Power Geometry for One ODE and P1 - P6 -- 1.1 Statement of the Problem -- 1.2 Computation of Truncated Equations -- 1.3 Computation of Expansions of Solutions to the Initial Equation (1.1) . -- 1.4 Extension of the Class of Solutions -- 1.5 Solution of Truncated Equations -- 1.6 Types of Expansions -- 1.7 Painlevé Equations Pl -- 2 New Simple Exact Solutions to Equation P6 -- 2.1 Introduction -- 2.1.1 Power Geometry Essentials -- 2.1.2 Matching "Heads" and "Tails" of Expansions -- 2.2 Constructing the Template of an Exact Solution -- 2.3 Results -- 2.3.1 Known Exact Solutions to P6 -- 2.3.2 Computed Solutions -- 2.3.3 Generalization of Computed Solutions -- 3 Convergence of a Formal Solution to an ODE -- 3.1 The General Case -- 3.2 The Case of Rational Power Exponents -- 3.3 The Case of Complex Power Exponents -- 3.4 On Solutions of the Sixth Painlevé Equation -- 4 Asymptotic Expansions and Forms of Solutions to P6 -- 4.1 Asymptotic Expansions near Singular Points of the Equation -- 4.2 Asymptotic Expansions near a Regular Point of the Equation -- 4.3 Boutroux-Type Elliptic Asymptotic Forms -- 5 Asymptotic Expansions of Solutions to P5 -- 5.1 Introduction -- 5.2 Asymptotic Expansions of Solutions near Infinity -- 5.3 Asymptotic Expansions of Solutions near Zero -- 5.4 Asymptotic Expansions of Solutions in the Neighborhood of the Nonsingular Point of an Equation -- II Space Power Geometry -- 6 Space Power Geometry for one ODE and P1 - P4, P6 -- 6.1 Space Power Geometry -- 6.2 Asymptotic Forms of Solutions to Painlevé Equations P1 - P4, P6 -- 6.2.1 Equation P1 -- 6.2.2 Equation P2 -- 6.2.3 Equation P3 for cd ≠ 0 -- 6.2.4 Equation P3 for c = 0 and ad ≠ 0 -- 6.2.5 Equation P3 for c = d = 0 and ab ≠ 0 -- 6.2.6 Equation P4 -- 6.2.7 Equation P6.

7 Elliptic and Periodic Asymptotic Forms of Solutions to P5 -- 7.1 The Fifth Painlevé Equation -- 7.2 The case δ ≠ 0 -- 7.2.1 General Properties of the P5 Equation -- 7.2.2 The First Family of Elliptic Asymptotic Forms -- 7.2.3 The First Family of Periodic Asymptotic Forms -- 7.2.4 The Second Family of Periodic Asymptotic Forms -- 7.3 The Case δ ≠ 0, γ ≠ 0 -- 7.3.1 General Properties -- 7.3.2 The Second Family of Elliptic Asymptotic Forms -- 7.3.3 The Third Family of Periodic Asymptotic Forms -- 7.3.4 The Fourth Family of Periodic Asymptotic Forms -- 7.4 The Results Obtained -- 8 Regular Asymptotic Expansions of Solutions to One ODE and P1-P5 -- 8.1 Introduction -- 8.2 Finding Asymptotic Forms -- 8.3 Computation of Expansions (8.2) -- 8.4 Equation P1 -- 8.5 Equation P2 -- 8.5.1 Elliptic Asymptotic Forms, Face Γ3(2) -- 8.5.2 Periodic Asymptotic Forms, Face Γ4(2) -- 8.6 Equation P3 -- 8.6.1 Case cd ≠ 0 -- 8.6.2 Case c = 0, ad ≠ 0 -- 8.6.3 Case c = d = 0, ab ≠ 0 -- 8.7 Equation P4 -- 8.7.1 Elliptic Asymptotic Forms, Face Γ3(2) -- 8.7.2 Periodic Asymptotic Forms, Face Γ4(2) -- 8.8 Equation P5 -- 8.8.1 Case d ≠ 0, Elliptic Asymptotic Forms, Face Γ1(2) -- 8.8.2 Case d ≠ 0, Periodic Asymptotic Forms, Face Γ2(2) -- 8.8.3 Case d = 0, c ≠ 0, Elliptic Asymptotic Forms, Face Γ1(2) -- 8.8.4 Case d = 0, c ≠ 0, Periodic Asymptotic Forms, Face Γ2(2) -- III Isomondromy Deformations -- 9 Isomonodromic Deformations on Riemann Surfaces -- 9.1 Introduction -- 9.2 The Space of Parameters T̃ -- 9.3 The Description of Bundles with Connections on a Riemann Surface -- 9.4 Isomonodromic Deformations -- 10 On Birational Darboux Coordinates of Isomonodromic Deformation Equations Phase Space -- 11 On the Malgrange Isomonodromic Deformations of Nonresonant Irregular Systems -- 11.1 Introduction -- 11.2 The Malgrange Isomonodromic Deformation of the Pair (E0, V̄0).

11.3 Specificity of Meromorphic 2x2-Connections -- 12 Critical behavior of P6 Functions from the Isomonodromy Deformations Approach -- 12.1 Introduction -- 12.2 Behavior of y(x) -- 12.3 Parameterization in Terms of Monodromy Data -- 13 Isomonodromy Deformation of the Heun Class Equation -- 13.1 Introduction -- 13.2 Gauge Transforms of Linear Differential Equations -- 13.3 Gauge Transforms of the Hypergeometric Class Equations -- 13.4 Gauge Transform of Heun Class Equations -- 13.4.1 Formulation of the Problem -- 13.4.2 Initial System of Equations and Equation Heunc2 -- 13.4.3 Parameters of the Transformed Equation -- 13.5 Conclusion -- 14 Isomonodromy Deformations and Hypergeometric-Type Systems -- 14.1 Schlesinger Families of Fuchsian Systems -- 14.2 Schlesinger Systems -- 14.3 Upper-Triangular Schlesinger Systems -- 14.4 Jordan-Pochhammer Systems -- 14.5 The Basic Result -- 15 A Monodromy Problem Connected with P6 -- 15.1 Preliminaries I -- 15.2 Preliminaries II -- 15.3 Main Result -- 15.4 Example -- 16 Monodromy Evolving Deformations and Confluent Halphen's Systems -- 16.1 Introduction -- 16.2 Quadratic Systems and Nonassociative Algebras -- 16.3 Monodromy Evolving Deformations -- 16.4 Halphen's Confluent Systems and Monodromy Evolving Deformations -- 17 On the Gauge Transformation of the Sixth Painlevé Equation -- 17.1 Linearizations of the Sixth Painlevé Equation -- 17.1.1 LODE LVI -- 17.1.2 LODE LVI -- 17.1.3 LODE LVI -- 17.1.4 Schlesinger System with Symmetric Gauge -- 17.1.5 Schlesinger System with Asymmetric Gauge -- 17.2 Schlesinger Transformation LVI → LVI -- 18 Expansions for Solutions of the Schlesinger Equation at a Singular Point -- 18.1 Introduction -- 18.2 Schlesinger Equation and Isomonodromic Deformations -- 18.3 Sketch of the Proof -- IV Painlevé Property.

19 Painleve Analysis of Lotka-Volterra Equations -- 20 Painlevé Test and Briot-Bouquet Systems -- 21 Solutions of the Chazy System -- 22 Third-Order Ordinary Differential Equations with the Painlevé Test -- 22.1 Introduction -- 22.2 Simplified Equation -- 22.3 Reduced Equations -- 22.3.1 Leading Order k =-1 -- 22.3.2 Leading Order k = -2 -- 22.3.3 Leading Order k = -3 -- 22.3.4 Leading Order k = -4 -- 23 Analytic Properties of Solutions of a Class of Third-Order Equations with an Irrational Right-Hand Side -- V Other Aspects -- 24 The Sixth Painlevé Transcendent and Uniformizable Orbifolds -- 24.1 Algebraic Solutions of P6 and Uniformization Theory -- 24.2 On the General Solution to Equation (24.1) -- 24.3 Calculus: Abelian Integrals and Affine (Analytic) Connections -- 25 On Uniformizable Representation for Abelian Integrals -- 25.1 Introduction -- 25.2 Schwarz Equation and Equations on Tori -- 25.3 Holomorphic Elliptic Integrals and Hypergeometric Functions -- 25.3.1 Lemniscate -- 25.3.2 Equi-Anharmonic Curve -- 25.4 Abelian Integrals for Genus g > 1 -- 25.4.1 Higher Genera. Examples -- 26 Phase Shift for a Special Solution to the Korteweg-de Vries Equation in the Whitham Zone -- 26.1 Introduction -- 26.2 Evaluation of the Phase Shift -- 27 Fuchsian Reduction of Differential Equations -- 27.1 Fuchsian Reduction -- 27.1.1 Two Simple Examples -- 27.1.2 A More Complex Example -- 27.2 Two Applications: Astronomy and Relativity Theory -- 27.2.1 Astronomy. A Model of Gaseous Stars -- 27.2.2 Relativity. Gowdy Space-Time -- 27.3 Fuchsian Systems for Feynman Integrals -- 28 The Voros Coefficient and the Parametric Stokes Phenomenon for the Second Painlevé Equation -- 28.1 Introduction -- 28.2 Connection Formula for the Parametric Stokes Phenomenon.

28.3 Derivation of the Connection Formulas Through the Analysis of the Voros Coefficient of (P2) -- 29 Integral Symmetry and the Deformed Hypergeometric Equation -- 30 Integral Symmetries for Confluent Heun Equations and Symmetries of Painleve Equation P5 -- 31 From the Tau Function of Painlevé P6 Equation to Moduli Spaces -- 32 On particular Solutions of q-Painlevé Equations and q-Hypergeometric Equations -- 32.1 Introduction -- 32.2 q-Difference Equation of the Hypergeometric Type -- 32.3 Hypergeometric Solutions of the q-Painlevé Equations -- 33 Derivation of Painlevé Equations by Antiquantization -- 34 Integral Transformation of Heun's Equation and Apparent Singularity -- 34.1 Heun's Equation and Integral Transformation -- 34.2 Apparent Singularity and Integral Representation of Solutions -- 34.3 Elliptical Representation of Heun's Equation and Integral Transformation -- 35 Painlevé Analysis of Solutions to Some Nonlinear Differential Equations and their Systems Associated with Models of the Random-Matrix Type -- 35.1 Introduction -- 35.2 Model of the Random-Matrix Type with Airy Kernal -- 35.3 System of Differential Equations Associated with the Dyson Process -- 35.4 Solutions of the Traveling-Wave Form of a Partial Differential Equation -- 36 Reductions on the Lattice and Painlevé Equations P2, P5, P6 -- 36.1 Introduction -- 36.2 Symmetries of the ABS Equations -- 36.3 Reduction on the Lattice and Discrete Painlevé Equations -- 36.4 Continuous Symmetry Reductions -- Comments.