A Lax pair representation of a nonlinear dynamical system is presented which emerges in linear programming as a continuous version of Karmarkar's projective scaling algorithm. Karmarkar's continuous trajectory is also shown to be a gradient system. An expression of solution in a rational form is found by integrating the associated dynamical system. Fixed points of the trajectory also studied.Key words: Karmarkar's projective sca/ing algorithm, Lax pair form, gradient system, Toda lattice w I n t r o d u c t i o n There has been considerable interest in studying nonlinear dynamical systems for solving linear programming problems through interior-point algorithms since Karmarkar's idea [10] turned such problems into a problem in nonlinear ODEs. We here consider three nonlinear dynamical systems in linear programming.An algorithm introduced by Dikin [6] in 1967 is nowadays called the aŸ scaling algorithm. It is pointed out in [10] that the associated dynamical system, a continuous version of the algorithm, is essentially a gradient system with respect to a generalized Poincar› metric. Bayer and Lagarias [2, 3] and Faybusovich [7] showed that the dynamical system can be interpreted a s a completely integrable Hamiltonian system. The equations of motion ave linearized by a Legendre transformation. The affine scaling algorithm is simpler than other interior-point algorithms, however, it is believed [13] that it is n o t a polynomial-time algorithm.Karmarkar [10], in 1984, proposed an epoch-making polynomial-time linear programming algorithm called Karmarkar's algorithm of the projective scaling algorithm. A geometric structure of the algorithm was also discussed in [2,3,11,12]. Ir was shown in [11] that the tangent vector of the associated continuous trajectory can be interpreted as that of a geodesic on the interior of the feasible solution polytope. An integrability of rector fields of interior-point algorithms was also discussed by Iri in [9].One of the most important notions in the recent developments in integrable dynamical systems is their representation as an isospectral deformation of a certain linear operator L(x(t)) depending on a formal indeterminate x(t). The representation automatically exhibits first integrals as invariants of L(x(t)). See, for example, [1, p.59]. The isospectral deformation serves to express the dynamical system in a Lax pair form