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Preface; Contents; Symbols; 1 The Fundamentals of the Theory of Integrability of Differential Systems; 1.1 Existence and Properties of the First Integrals; 1.1.1 Characterization and Properties of the First Integrals; 1.1.2 Existence of First Integrals Near a Regular Point; 1.2 First Integrals of Differential Systems in Canonical Regions; 1.3 Applications of Integrability Theory to Partial Differential Equations; 1.3.1 First-Order Linear Homogeneous Partial Differential Equations; 1.3.2 First-Order Quasilinear Partial Differential Equations; 1.4 Lax Pairs and Integrability
2 Jacobian and Inverse Jacobian Multipliers2.1 Jacobian Multipliers, First Integrals and Integrability; 2.2 Inverse Jacobian Multipliers and Their Vanishing Sets; 2.3 Inverse Jacobian Multipliers and the Center-Focus Problem; 2.3.1 The Center-Focus Problem via Inverse Integrating Factors or Inverse Jacobian Multipliers; 2.3.2 Hopf Bifurcation via Inverse Jacobian Multipliers; 2.4 Inverse Jacobian Multipliers via Lie Groups; 3 Darboux and Liouvillian Integrability; 3.1 The Classical Darboux Theory of Integrability; 3.1.1 The Existence of Darboux First Integrals
3.1.2 The Darboux
Jouanolou Integrability Theorem3.2 Generalization of the Classical Darboux Theory of Integrability; 3.2.1 Taking into Account Independent Singularities; 3.2.2 Taking into Account Algebraic Multiplicities; 3.2.3 Taking into Account the Multiplicity of the Hyperplane at Infinity; 3.2.4 On Nonautonomous Differential Systems via the Wronskian Matrix; 3.2.5 Differential Systems in the Sparse Case; 3.2.6 Other Extensions; 3.3 Liouville and Elementary First Integrals; 3.3.1 Background on Differential Field Extensions; 3.3.2 The Prelle and Singer Integrability Theorems
3.4 Liouvillian Integrability Versus Darboux Polynomials4 Existence and Degree of Darboux Polynomials; 4.1 The Degree of Invariant Algebraic Curves; 4.1.1 Examples of Invariant Algebraic Curves of Arbitrary Degree; 4.1.2 Invariant Algebraic Curves in the Projective Plane; 4.1.3 The Degree of Invariant Algebraic Curves in the Nodal and Nondicritical Cases; 4.2 Existence of Darboux Polynomials; 4.2.1 Polynomial Vector Fields Without Darboux Polynomials; 4.2.2 Liénard Differential Systems: Invariant Algebraic Curves; 4.2.3 Lorenz Systems: Invariant Algebraic Surfaces
4.3 Other Results on Darboux Polynomials5 Algebraic, Analytic and Meromorphic Integrability; 5.1 Algebraic First Integrals; 5.1.1 Algebraic and Rational Integrability: Their Equivalence; 5.1.2 Kirchoff Equations: Polynomial and Rational First Integrals; 5.1.3 Euler Equations on the Lie Algebra so(4): Polynomial First Integrals; 5.1.4 The 5-Dimensional Lorenz Systems: Darboux and Analytic Integrability; 5.2 Natural Hamiltonian Systems: Polynomial and Rational Integrability; 5.2.1 Hamiltonian Systems in the Canonical Form; 5.2.2 Hamiltonian Systems in a Generalized Form
2 Jacobian and Inverse Jacobian Multipliers2.1 Jacobian Multipliers, First Integrals and Integrability; 2.2 Inverse Jacobian Multipliers and Their Vanishing Sets; 2.3 Inverse Jacobian Multipliers and the Center-Focus Problem; 2.3.1 The Center-Focus Problem via Inverse Integrating Factors or Inverse Jacobian Multipliers; 2.3.2 Hopf Bifurcation via Inverse Jacobian Multipliers; 2.4 Inverse Jacobian Multipliers via Lie Groups; 3 Darboux and Liouvillian Integrability; 3.1 The Classical Darboux Theory of Integrability; 3.1.1 The Existence of Darboux First Integrals
3.1.2 The Darboux
Jouanolou Integrability Theorem3.2 Generalization of the Classical Darboux Theory of Integrability; 3.2.1 Taking into Account Independent Singularities; 3.2.2 Taking into Account Algebraic Multiplicities; 3.2.3 Taking into Account the Multiplicity of the Hyperplane at Infinity; 3.2.4 On Nonautonomous Differential Systems via the Wronskian Matrix; 3.2.5 Differential Systems in the Sparse Case; 3.2.6 Other Extensions; 3.3 Liouville and Elementary First Integrals; 3.3.1 Background on Differential Field Extensions; 3.3.2 The Prelle and Singer Integrability Theorems
3.4 Liouvillian Integrability Versus Darboux Polynomials4 Existence and Degree of Darboux Polynomials; 4.1 The Degree of Invariant Algebraic Curves; 4.1.1 Examples of Invariant Algebraic Curves of Arbitrary Degree; 4.1.2 Invariant Algebraic Curves in the Projective Plane; 4.1.3 The Degree of Invariant Algebraic Curves in the Nodal and Nondicritical Cases; 4.2 Existence of Darboux Polynomials; 4.2.1 Polynomial Vector Fields Without Darboux Polynomials; 4.2.2 Liénard Differential Systems: Invariant Algebraic Curves; 4.2.3 Lorenz Systems: Invariant Algebraic Surfaces
4.3 Other Results on Darboux Polynomials5 Algebraic, Analytic and Meromorphic Integrability; 5.1 Algebraic First Integrals; 5.1.1 Algebraic and Rational Integrability: Their Equivalence; 5.1.2 Kirchoff Equations: Polynomial and Rational First Integrals; 5.1.3 Euler Equations on the Lie Algebra so(4): Polynomial First Integrals; 5.1.4 The 5-Dimensional Lorenz Systems: Darboux and Analytic Integrability; 5.2 Natural Hamiltonian Systems: Polynomial and Rational Integrability; 5.2.1 Hamiltonian Systems in the Canonical Form; 5.2.2 Hamiltonian Systems in a Generalized Form