  ( We present complete solutions of $K$-matrix for the quantum
Mikhailov-Shabat model. It has been known that there are three diagonal
solutions with no free parameters, one being trivial identity solution, the
others non-trivial. The most general solutions which we found consist of three
families corresponding to each diagonal solutions. One family of solutions
depends on two arbitrary parameters. If one of the parameters vanishes, the
other must also vanish so that the solutions reduces to trivial identity
solution. The other two families for each non-trivial diagonal solutions have
only one arbitrary parameter.)
