  We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by
a weaker condition which we call "rime". Rime solutions include the standard
Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime
Ansatz which are maximally different from the standard one we call "strict
rime". A strict rime non-unitary solution is parameterized by a projective
vector. We show that this solution transforms to the Cremmer-Gervais R-matrix
by a change of basis with a matrix containing symmetric functions in the
components of the parameterizing vector. A strict unitary solution (the rime
Ansatz is well adapted for taking a unitary limit) is shown to be equivalent to
a quantization of a classical "boundary" r-matrix of Gerstenhaber and
Giaquinto. We analyze the structure of the elementary rime blocks and find, as
a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly
described in a rime form.
  We discuss then connections of the classical rime solutions with the Bezout
operators. The Bezout operators satisfy the (non-)homogeneous associative
classical Yang--Baxter equation which is related to the Rota-Baxter operators.
  We classify the rime Poisson brackets: they form a 3-dimensional pencil. A
normal form of each individual member of the pencil depends on the discriminant
of a certain quadratic polynomial. We also classify orderable quadratic rime
associative algebras.
  For the standard Drinfeld-Jimbo solution, there is a choice of the
multiparameters, for which it can be non-trivially rimed. However, not every
Belavin-Drinfeld triple admits a choice of the multiparameters for which it can
be rimed. We give a minimal example.
