  In recent work, we presented the construction of a family of difference
equations associated with the Stieltjes continued fraction expansion of a
certain function on a hyperelliptic curve of genus $g$. As well as proving that
each such discrete system is an integrable map in the Liouville sense, we also
showed it to be an algebraic completely integrable system. In the discrete
setting, the latter means that the generic level set of the invariants is an
affine part of an abelian variety, in this case the Jacobian of the
hyperelliptic curve, and each iteration of the map corresponds to a translation
by a fixed vector on the Jacobian. In addition, we demonstrated that, by
combining the discrete integrable dynamics with the flow of one of the
commuting Hamiltonian vector fields, these maps provide genus $g$
algebro-geometric solutions of the infinite Volterra lattice, which justified
naming them Volterra maps, denoted ${\cal V}_g$.
  The original motivation behind our work was the fact that, in the particular
case $g=2$, we could recover an example of an integrable symplectic map in four
dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified
birational maps in 4D admitting two invariants (first integrals) with a
particular degree structure, by considering recurrences of fourth order with a
certain symmetry. Hence, in this particular case, the map ${\cal V}_2$ yields
genus two solutions of the Volterra lattice. The purpose of this note is to
point out how two of the other 4D integrable maps obtained in the
classification of Gubbiotti et al. correspond to genus two solutions of two
different forms of the modified Volterra lattice, being related via a
Miura-type transformation to the $g=2$ Volterra map ${\cal V}_2$.
  We dedicate this work to a dear friend and colleague, Decio Levi.
