  We formulate an analog of Inverse Scattering Method for integrable systems on
noncommutative associative algebras. In particular we define Hamilton flows,
Casimir elements and noncommutative analog of the Lax matrix. The
noncommutative Lax element generates infinite family of commuting Hamilton
flows on an associative algebra. The proposed approach to integrable systems on
associative algebras satisfy certain universal property, in particular it
incorporates both classical and quantum integrable systems as well as provides
a basis for further generalization.
  We motivate our definition by explicit construction of noncommutative analog
of Lax matrix for a system of differential equations on associative algebra
recently proposed by Kontsevich. First we present these equations in the
Hamilton form by defining a bracket of Loday type on the group algebra of the
free group with two generators. To make the definition more constructive we
utilize (with certain generalizations) the Van den Bergh approach to Loday
brackets via double Poisson brackets. We show that there exists an infinite
family of commuting flows generated by the noncommutative Lax element.
