Painlevé property and conservation laws of multi-component mKdV equations

It is shown that two classes of n-component mKdV equations are Painlevé integrable, and the resonances occur, respectively, at −1,3,4, , , for the geometric n-component mKdV equation which arises from curve motions in Euclidean space, and −1,3,4, , , for another n-component mKdV equation which has an infinite number of Lie Bäcklund symmetries. It is also shown that the two-component geometric mKdV equation admits an infinite number of conservation laws. Conservation laws for several systems are constructed.