Recently the third named author defined a 2-parametric family of groups Gkn . Those groups may be regarded as a certain generalisation of braid groups. Study of the connection between the groups Gkn 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 Gkn. The Gkn groups have connections to different algebraic structures, Coxeter groups and Kirillov–Fomin algebras, to name just a few. Study of the Gkn 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 Gkk+1 groups are algorithmically solvable.