In [OOS] we announce a classification of integrable systems invariant under simple classical Weyl groups. The precise discussion has already been given by [OS] and [O] except for the case of type . As is shown in [OS], the classification problem for type is reduced to a functional differential equation (1.4).

In §2 we give a complete list of solutions of this functional equation. Some solutions have already been obtained, after [OP2], by Inozemtsev [IM], [I] (See also [P]). The main result of §2 is Theorem 2.9, which is stated in §1.3 in a different form.

In §3 we examine the reducibility of the system obtained in §2. We note that if the system coincides with the system satisfied by zonal spherical functions of a semisimple Lie group, the reducibility is related to degenerate series representations.

The final draft of this paper was completed when the authors were visiting University of Leiden in the fall of 1994. The authors express their sincere gratitude to Profdrvan Dijk for his hospitality during their stay there.

**1.2.**
Now we give a quick review of the results in [OS, §6]
concerning with type .
Let be the Weyl group of type , which is identified with
the group of coordinate transformations of generated by
and
.
Consider -invariant differential operators

which satisfies and . Here we denote and for simplicity and the map is the anti-automorphism of the algebra of differential operators such that for functions and for and 2. We assume that the coefficients of differential operators are extended to holomorphic functions on a Zariski open subset of an open connected neighborhood of the origin of the complexification of .

The operators are proved to be expressed by even functions and of one variable as follows ([OS, Proposition 6.3]):

where is determined by the following equations up to a constant.

As the compatibility condition for the existence of the solution of the equation (1.3), we have an equation

which have been posed in [OS, Proposition 6.3] (cf. [P, §2.2.C]).

Conversely for any solution of (1.4) and the pair of the operators which are given by (1.2) with

under the notation in Remark 2.1 and Lemma 2.2, we have .

**1.3.**
We give a complete list of solutions of the functional equation (1.4).
Remind that the Schrödinger operator is explicitly expressed
as in (1.2) using and .

(Trivial case) constant, an arbitrary even function,

an arbitrary even function, constant.

Let and denote the primitive half periods of the Weierstrass elliptic function and put and .

(Elliptic case) For ,

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(Trigonometric case)
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(Rational case)
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(Elliptic case) For ,
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(Trigonometric case)
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(Rational case)
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**1.4.**
Although we deal with the commuting differential operators of type
with the Weyl group symmetry
in the main body of this paper,
we will give a brief summary of the related works.

The commuting differential operators of type have been studied very well. The commuting differential operators of type with the Weyl group invariant condition are classified in [OS]. This work is generalized to the commuting differential operators of type without Weyl group invariant condition

For the Weyl group invariant case, we have and the proof of this fact is given in Proposition 4.2 (with ) of [OS], which is valid for the general case with no change. For the Weyl group invariant case, the functional differential equation (1.6) is solved in [WW] and the solution is a Weierstrass elliptic function . The corresponding potential is of Calogero-Moser type. For the general case, the equation (1.6) is solved in [BP] and [BB]. Besides the solutions, we also have solutions expressed by exponential functions. The corresponding potential is known as of type periodic/non-periodic Toda, which can be regarded as a degenerating limit of a Weyl group invariant potential [vD].

For type , the classification of the commuting differential operators (1.1) without the Weyl group symmetry has not been done yet. It is known that the similar functional differential equation (see (2.3)) is related to such operators. The following results are obtained in [Oc]:

(i) We have the expression of the (non Weyl group invariant) operators and by using four functions , , and with one-variable. Actually, if we replace by , by , and so on, the formula (1.2) is also valid for non-invariant operators. These functions satisfy the functional differential equation like (1.4).

(ii) Suppose be non-trivial (cf. Lemma 2.4 i)). If is holomorphic at some point, then and can be meromorphically continued to whole plane . The orders of poles of are at most two.

(iii) Suppose, moreover, that has poles at three points such that and are linearly independent over . Then the function can be expressed as