  A family of nonparametric Yang Baxter (YB) maps is constructed by
refactorization of the product of two 2 by 2 matrix polynomials of first
degree. These maps are Poisson with respect to the Sklyanin bracket. For each
Casimir function a parametric Poisson YB map is generated by reduction on the
corresponding level set. By considering a complete set of Casimir functions
symplectic multiparametric YB maps are derived. These maps are quadrirational
with explicit formulae in terms of matrix operations. Their Lax matrices are,
by construction, 2 by 2 first degree polynomial in the spectral parameter and
are classified by Jordan normal form of the leading term. Nonquadrirational
parametric YB maps constructed as limits of the quadrirational ones are
connected to known integrable systems on quad graphs.
