By Mark Adler

From the reports of the 1st edition:

"The target of this e-book is to give an explanation for ‘how algebraic geometry, Lie conception and Painlevé research can be utilized to explicitly resolve integrable differential equations’. … one of many major merits of this e-book is that the authors … succeeded to offer the cloth in a self-contained demeanour with a number of examples. therefore it may be extensively utilized as a reference publication for plenty of matters in arithmetic. In precis … an outstanding booklet which covers many attention-grabbing matters in glossy mathematical physics." (Vladimir Mangazeev, The Australian Mathematical Society Gazette, Vol. 33 (4), 2006)

"This is an intensive quantity dedicated to the integrability of nonlinear Hamiltonian differential equations. The e-book is designed as a educating textbook and goals at a large readership of mathematicians and physicists, graduate scholars and pros. … The ebook offers many beneficial instruments and methods within the box of thoroughly integrable structures. it's a necessary resource for graduate scholars and researchers who prefer to input the integrability conception or to benefit attention-grabbing features of integrable geometry of nonlinear differential equations." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1083, 2006)

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**Additional info for Algebraic Integrability, Painlevé Geometry and Lie Algebras**

**Example text**

Let (M,{·,·}) be a Poisson manifold. The (singular) distribution on M defined by the Hamiltonian vector fields is integrable in the sense that every m E M has a coordinate neighborhood U which is, in a unique way, a disjoint union of symplectic manifolds Ui which are Poisson submanifolds of U. The resulting (singular) foliation is called the symplectic foliation and each of its leaves is called a symplectic leaf. Proof. For m E M, let U denote a coordinate neighborhood of m with coordinates (qt, ...

When v = Id 9 then 34 2 Lie Algebras L(g) := L (g,Id 9 ) = g ® C [b,b- 1] the affine Lie algebra of g. The term loop algebra is also used. A natural Lie bracket on L(g, v) is given by and the Killing form (·I·) on g leads for every k E Z to a non-degenerate symmetric form on L(g, v), denoted by (·I· )k which is defined by It is easy to see that each of the bilinear forms (·I· )k on L(g, v) is Adinvariant. We will refer to (·I· )o as the Killing form of L(g, v). 18. Consider the direct sum decomposition L(g) = L(g)+ Ee L(g)_, where L(g)+ consists of those elements of L(g) which are polynomial in b, while L(g)_ consists of all elements of L(g) that are polynomial in b- 1 , but without constant term.

Picking local coordinates (x1, ... , Xn) we can introduce a skew-symmetric n x n matrix {} by nii := w ( a~i, a~j) . We claim that the Poisson matrix X of {· , ·} with respect to these coordinates is given by X = -n- 1 • To show this, let us denote by [XF] the column matrix whose elements are the coefficients of XF with respect to the basis ( 8~ 1 , ••• , 8 ~n) and let us recall that we denote by [dF] the column matrix whose elements are the coefficients of dF with respect to the basis {dx1, ...