Now we recall the definition of the first Painlevé hierarchy. Consider the serial equations

for, where is an expression of a given meromorphic function defined by and with. The equations are derived from the singular manifold equation for the KdV hierarchy, and we call them the first Painlevé hierarchy [10] [12] [13]. For example, , where is an integral constant. In the followings, each is also an integral constant.

If, we have then essentially coinsides with the first Painlevé equation.

If, we have and then

Again, it essentially coinsides with (5), i.e. the 4th order equation of the first Painlevé hierarchy.

As proved by K. Takasaki [8], the string equation of type is equivalent to (6). So, in this article, we also call (6) the string equation of type.

Note that S. Shimomura [14] proved the theorems as follows.

Theorem A [14]. Each is a differential polynomial of 2n-th order, i.e. each (6) is an ordinary differential equation of 2n-th order.

Theorem B [14]. At each pole, the meromorphic solution to (6) has the form

for some positive integer.

The author proved a theorem similar to Theorem A for the second Painlevé hierarchy [15], and, in its proof, auxiliary differential polynomials play important roles. So, for the first Painlevé hierarchy as well, the auxiliary differential polynomials should exist. Recall them.

Theorem C [16]. Define by. Then each is a differential polynomial of 2n-th order.

The first Painlevé equation has the autonomous limit [9]. Replacing by with a constant, and taking limit, we obtain which is satisfied by the Weierstrass’ elliptic function, i.e.

where for some satisfying. Here means the sum for. It is well-known that is a doubly periodic meromorphic function with two fundamental periods. satisfies the differential equation

and then

,

where

For, a similar result is valid, i.e.

Theorem D [17]. The 4th order equation of the first Painlevé hierarchy with suitable parameters

is reduced to the autonomous equation

by replacing by, and taking the limit.

Note that the Equation (8) is obtained as a section of the most degenerated 2D Garnier system [18] (see also [19] [20]). The following theorem is not trivial but natural if we consider Theorem D together with Theorem B.

Theorem E [17]. For suitable, the autonomous Equation (9) has a solution concretely described as

A result similar to Theorem D is valid for.

Theorem 1.1. The autonomous limit of the string equation of type is given by

where is a complex parameter.

Proof. Note that has the form

with the weight [14] defined by and. After replacement

,

taking the limit, we obtain the conclusion.

For the autonomous limit Equation (10), each auxiliary differential polynomial obtained in Theorem C has clear meaning.

Theorem 1.2. The differential polynomial is the first integral of (10).

Proof. By definition,. Using this relation together with the equation,we obtain

Now we extend Theorem E to the case where

Theorem 1.3. Weierstrass’ elliptic function is a solution to (10) with suitable parameters.

Moreover, we can prove the theorem as follows:

Theorem 1.4. For each integer satisfying,

is a solution to (10) with suitable parameters.

The proofs of these two theorems are given in the next section.

Let all of’s for vanish. Take satisfying

with suitable, then we have

with suitable’s. Putting, i.e. choosing as it satisfies,we obtain the conclusion.

Theorem 1.4 immediately follows from the following lemma.

Lemma. For every positive integer, is described by some polynomial of, and its degree is as follows:

Proof. Using and,

So, if, the coefficient of vanishes as follows:

Thus, we have

Note that is a polynomial in of degree, and all terms but one of top degree have integral constants. Therefore, if the term of top degree vanishes, we can make all terms vanish with suitable selection of integral constants. Thus, Theorem 1.4 is established.