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The $KIO3$.$HIO3$ crystals belong to the family of acidic iodate compounds which is well-known for its ferroelectric phase transitions and has four crystal modifications. Here we report the results of IR and Raman study of vibrational spectra ($4000-60cm-1$) of $\alpha$ and $\sigma$-modifications in the temperature range from 300 to 13 K. The $\alpha -KIO3$ $HIO3$ crystal ($P21/b$ space group, z=4 at room temperature) is built of two nonequivalent sets of dimers $IO3\cdots H\cdots O3I$, joined by strong symmetric hydrogen bond (O...O distance is about 2.5 \AA). At 223 K ($Te$) the ferroelectric phase transition occurs in this crystal. The stable $\sigma$-modification (space group $Fdd2,z=24$ at room temperature) is formed from structure units of three distinct types:$I2O5(OH)$, which are joined by the H-bound network. In order to understand the normal modes of vibrations in $\alpha -KIO3-HIO3$.and $\sigma -KIO3\cdot HIO3$, we have carried out lattice dynamics calculations using a generalized valence force field for the dimers $[H(IO3)2]-$ and othe structure units. The introduction of the translation and libration coordinates related to the normal coordinate analysis leads to a precise assignment of the frequencies and shows that both stretching and bending vibrations of $IO3$ groups are strongly coupled with the translational vibrations of the hydrogen atom in case of $\sigma$-modification. In the case of $\sigma$-modification those vibrations are localized on separate I-O bonds. A comparative analysis of the vibrational spectra and structure is presented. On the base of analysis of vibrational spectra and according to $measurementsa$ of pressure dependencies of the $\mathrm{<mrow\; data-mjx-texclass="ORD">127</mrow>}I$ NQR spectra at 77 K it was concluded that the mechanism of ferroelectric phase transition in $\alpha$-modification of potassium biiodate crystal involves the proton ordering in the double-well potential at $Tc$.