  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 [AS19] 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 $SL(2,\mathbb C)$-charater 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.
