Abstract. In this work we generalise previous results connecting (rational) Gaudin magnet models and classical separation of variables. It is shown that the connection persists for the case of linear r-matrix algebra which corresponds to the trigonometric

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E.G. KALNINS, V.B. KUZNETSOV AND WILLARD MILLER, JR.

recall the fundamental ideas of the r-matrix formalism (see 10,11] and references in there). For a classical mechanical system the basic (linear) r-matrix algebra is (1.1)

fL(u) I; I L(v)g= r(u? v); L(u) I+ I L(v)];L(u)11=?L(u)22= A(u); L(u)12= B (u); L(u)21= C (u);

where f; g is the Poisson bracket and;] the matrix commutator bracket. The operator L(u) is taken to be the 2 2 matrix and r(u) is a suitable 4 4 matrix of scalars solving the classical Yang-Baxter equation 10,11]; u being arbitrary constant is called the spectral parameter. In the case of the XXZ r-matrix algebra the non zero elements of r can be taken to be 1 (1.2) r(u)11= r(u)44= coth(u); r(u)23= r(u)32= sinh(u): In component form, the r-matrix algebra relations are (1.3)

1 fA(u); C (v)g= sinh(u? v) (? cosh(u? v)C (v)+ C (u));? fB (u); C (v)g= sinh(u2? v) (A(u)? A(v)): If we now ma

ke the ansatz A(u)= coth(u)S3; B (u)= (1= sinh(u))S+ and C (u)= (1= sinh(u))S? these relations imply (1.4)n X

fA(u); A(v)g= fB (u); B (v)g= fC (u); C (v)g= 0; 1 fA(u); B (v)g= sinh(u? v) (cosh(u? v)B (v)? B (u));

fS3; S g= S;

fS+; S?g= 2S3:n X

To relate this observation to the separation of variables methods, we form the L(u) operator with elements 1 B (u)= sinh(u? e ) S+;=1

C (u)=

(1.5) where (1.6) (1.7)

1 sinh(u? e ) S?;=1

A(u)=

n X=1

coth(u? e ) S3;

fS3; S g=

S;

fS+; S? g= 2 S3:

The r-matrix algebra relations, (1.1) or (1.3), imply

fdet L(u); det L(v)g= 0;