### Abstract

We examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, autonomous second order ordinary differential equation (ODE) for m/R (m is the mass inside the radius R) that has been solved analytically only for n = −1 and n = −3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = −5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here, we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent autonomous Lotka–Volterra quadratic polynomial differential system in , and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka–Volterra system possesses Liouvillian first integrals for n = −1, −3, −5, which descend from the existence of invariant algebraic curves of degree one, and for n = −6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka–Volterra system, and consequently of the second order ODE for the mass function must be non-Liouvillian. This makes the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric quite unlikely.

Received 11 February 2014
Accepted 22 September 2014
Published online 07 October 2014

Acknowledgments:
The second author is partially supported by the MINECO/FEDER Grant No. MTM2008–03437, AGAUR Grant No. 2009SGR-410, ICREA Academia, FP7-PEOPLE-2012-IRSES 316338 and 318999, and Grant No. FEDER/UNAB10-4E-378.

Article outline:

I. INTRODUCTION AND STATEMENT OF THE MAIN RESULT
II. DERIVATION OF THE LOTKA–VOLTERRA SYSTEM (2)
III. BASIC RESULTS FROM THE DARBOUX THEORY OF INTEGRABILITY
IV. THE INVARIANT ALGEBRAIC CURVES OF SYSTEM (2)
V. THE EXPONENTIAL FACTORS OF SYSTEM (2)

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