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

10

E.G. KALNINS, V.B. KUZNETSOV AND WILLARD MILLER, JR.2 1 2 1 H1= k2 sn(e2? e ) (Z1 )1 (Z2 )1? k2cn(e1? e3 )(Z1 )3(Z2 )3 1 3 k2 cn(e1? e3 )dn(e1? e3 ) (Z 2 ) (Z 1 )+ dn(e1? e3 ) (Z 2) (Z 1);? 1 1 1 1 sn(e1? e3 ) sn(e1? e3 ) 1 3 1 3 H2=?H1;? 1 E (e e 1 2 2 2 H3= k2 sn(1?? 3e) ) H1+ k2 (Z1 )2? (Z2 )2? (Z1 )2? (Z1 )2? (Z1 )2 1 2 2 2 3 e1 3 2 1 2 1? 2k2 sn(e1? e3 )(Z1 )1 (Z1 )1+ 2dn(e1? e3 )(Z1 )3(Z2 )3; 1 1 1 1 H4= Z1:Z1; H5= 2Z1:Z2; 1 1 1 1 1 2 1 H6= Z2:Z2? (Z1 )2? (Z1 )2? k2(Z1 )2+ sn(e 2? e ) (Z1 )1(Z1 )1 1 2 3 1 3 cn(e1? e3 ) (Z 2) (Z 1);? 2 sn(e? e ) 1 3 2 3 1 3 2 2 H7= Z1:Z1:

where

We note that the ideas developed here also work in the case of separation of variables for spaces of constant Riemannian curvature, as developed in previous articles 6{8]. Indeed, in that case the rational r-matrix algebra is as before and the non zero elements of the r-matrix are (3.7)

r(u)11= r(u)44= r(u)23= r(u)32= 1:

The generating function of the constants of the motion for signature N1;::::; Np is then (3.8)

0 p NJ? X XX det L(u)=@3

1

k=1 J=1 j=0

(ZjJ+1 )k A (u? J e1 )NJ?j+ k

1

2

:

This is the generalisation of the generating function for separable coordinates on P spaces of constant curvature of dimension n= p=1 NJ+ 1. Indeed, if we use the j form (3.8) and if k= 0 for k= 1; 2; 3 then we have the generating function on the sphere for generic ellipsoidal coordinates, and if 1=?1=4, 2= 1=4, 3= 0 then we have the generating function of ellipsoidal coordinates in n-dimensional Euclidean space. As an example consider the system with signature 2,1. The generating function is thenZ:Z det L(u)= e1?e3:: 1) (? ((Z1?:eZ)22) (Z11 )2(Z21 )? (Ze121)?(eZ311 ) e 3+ u? e1 (u? e1 )2 1 (Z1 ):(Z 1 ) (Z 2 ):(Z 1 ) 1 1 2 2 1 2 (Z1 ):(Z1 )+ (Z1 ):(Z1 )? e11?e32? (e1?e3 )22+ (Z1 ):(Z1 ):+ (u? e )3 (u? e )4 u? e3 (u? e3 )2 1 12 1 ( 1) ( 2) 1 1