  Recently the third named author defined a 2-parametric family of groups
$G_n^k$ \cite{gnk}. Those groups may be regarded as a certain generalisation of
braid groups. Study of the connection between the groups $G_n^k$ and dynamical
systems led to the discovery of the following fundamental principle: `If
dynamical systems describing the motion of $n$ particles possess a nice
codimension one property governed by exactly $k$ particles, then these
dynamical systems admit a topological invariant valued in $G_{n}^{k}$'.
  The $G_n^k$ groups have connections to different algebraic structures,
Coxeter groups and Kirillov-Fomin algebras, to name just a few. Study of the
$G_n^k$ groups led to, in particular, the construction of invariants, valued in
free products of cyclic groups.
  In the present paper we prove that word and conjugacy problems for certain
$G_{k+1}^k$ groups are algorithmically solvable, and the algorithms are
constructive.
