We show that one-parameter deformation \(\mathcal A_{q,t}\) of the skein algebra \(Sk_q(\Sigma _2)\) of a genus two surface suggested in Arthamonov and Shakirov (Sel Math New Ser 25(2):17, 2019) is flat. We solve the word problem in the algebra and describe monomial basis. In addition, we calculate the classical limit \(\mathcal A_{q=1,t}\) of the algebra and prove that it is a one-parameter flat Poisson deformation of the coordinate ring \(\mathcal A_{q=t=1}\) of an \(SL(2,\mathbb C)\) -character variety of a genus two surface. As a byproduct, we obtain a remarkably simple presentation in terms of generators and relations for the coordinate ring \(\mathcal A_{q=t=1}\) of a genus two character variety.
