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

6

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

with obvious separation equations. (We require J i= 0 and take the limit as the 0 J 1 ! 0 for h= 1;:::; NJ? 1, se

e 8].) The generating function for the constants h can be derived by applying these procedures to det L(u). We will, however, adopt a di erent and more general strategy. If we leave the matrix elements of L(u) in the form (1.5) and subject the resulting expression for det L(U ) 1? 2 det L(U )= 2 (2.13) where U= eu, and S+n 1 X=1

S3

!

2

n X S

3 U 2? A2?=1

2 n X X A Si !+ 2U 4 U?A i n n X X S !#2 2 2=1=1 2 2

S3+ U 2

=1

U 2? A2=1

3

;

= S1

iS2, to the transformations j= 1;:::; NJ; J= 1;:::; p; k= 0;:::; NJ;

AJ ! AJ+ J 1?1; j 1 j

NJ J S1 k0+ X(J 1 )k (J Sj )= Z J j?1 NJ?k; j=2

then we arrive at a general expression for the generating function det L(U ). The constants of the motion are obtained by the usual means of expanding the expression following from (2.13) in partial fractions and reading o the independant components. In the case of degenerate roots the expression can be readily modi ed. Accordingly we havep X J 1? 2 det L(U )= 1 2 J=1(ZNJ )3 22 2=1 1

!

2

(2.14)

+

0 p J? X@X NX 1+ 2U 6 4 i J j j! p J? X NX 1@ j1=1=0

2 !@ j Aj J (ZNJ?j )i A 2? A2@Aj U j

1

@Aj U? Aj J=1 j=0 j ! 0 p NJ?1 1 X@ J X?(ZNJ )3+ U 2 j=0 j ! J=12

1

!

2

J (ZNJ?j )3 j2

@@Aj

U? A2 j

1

!

13 J (ZNJ?j ) A5:3

From this expression constants of the motion can be deduced just as before. The separation of variables proceeds as usual in the case of the choice of coordinates as given in 8]. The expressions for the coordinates corresponding to multiple roots with signature N1; N2;:::; Np can be obtained from the generic case by the limiting procedures already outlined. In rational form the generic coordinates are (2.15)6=? x2= n=1(A2? Ui2 )] n=11Us] (kA2i AkA2 ): i j j s`6=i`? i