An explicit formula for the Hilbert symbol for Honda groups in a multidimensional local field

Based on the pairing on Cartier curves explicitly constructed in the previous paper of the authors, an explicit formula for the Hilbert symbol is constructed in a multidimensional local field of characteristic zero with residue field of positive characteristic on the formal module of a one-dimensional Honda formal group. In the proof a Shafarevich basis on the formal module is constructed, and so-called integer $\mu$-modules in two-dimensional local rings of a special form ( $\mu$-rings) are studied.