Completely regular semigroups are semigroups that are unions of their subgroups. They may be regarded as universal algebras with an associative binary multiplication and unary inversion. In this language they form a variety \({\mathcal C}{\mathcal R}\). We present subvarieties of \({\mathcal C}{\mathcal R}\) by the corresponding fully invariant congruences on the free unary semigroup U. We give some constructions of how to obtain new f.i. congruences from the given ones. The basic construction is the following: given an f.i. congruence \(\sim\) on U, put a\({\bar \sim}b\) iff \(E(a)=E(b)\), \(a\sim b\), O(a)\({\bar \sim}O(b)\), \(\ell (a){\bar \sim}\ell (b)\) (E(a) denotes the set of all variables of a, O(a) is the longest initial segment of a in all but one variable with removed non-matched ''('', \(\ell (a)\) is defined dually). For \({\mathcal V},{\mathcal W}\in {\mathcal L}({\mathcal C}{\mathcal R})\) (the lattice of all varieties of c.r. semigroups), put \({\mathcal V}\rho {\mathcal W}\) iff \(\overline{\sim_{{\mathcal V}}}=\overline{\sim_{{\mathcal W}}}\) for the corresponding f.i. congruences \(\sim_{{\mathcal V}}\) and \(\sim_{{\mathcal W}}\). The relation is a complete lattice congruence on \({\mathcal L}({\mathcal C}{\mathcal R})\). (The smallest \(\rho\)-class is formed by all varieties of bands, another \(\rho\)-class is the interval [the class of all groups, the class of all orthodox c.r. semigroups].) We characterize a certain part of \({\mathcal L}({\mathcal C}{\mathcal R})/\rho\). The following aim is partially reached: a reduction of the description of \({\mathcal L}({\mathcal C}{\mathcal R})\) to the description of \({\mathcal L}({\mathcal C}{\mathcal R})/\rho\).