  In 1883, Darboux gave a local classification of isothermic surfaces with one
family of planar curvature lines using complex analytic methods. His choice of
real reduction cannot contain tori. We classify isothermic tori with one family
of planar curvature lines. They are found in the second real reduction of
Darboux's description. We give explicit theta function formulas for the family
of plane curves. These curves are particular area constrained hyperbolic
elastica. With a Euclidean gauge, the Euler--Lagrange equation is lower order
than expected. In our companion paper (arXiv:2110.06335) we use such isothermic
tori to construct the first examples of compact Bonnet pairs: two isometric
tori related by a mean curvature preserving isometry. They are also the first
pair of isometric compact immersions that are analytic. Additionally, we study
the finite dimensional moduli space characterizing when the second family of
curvature lines is spherical. Isothermic tori with planar and spherical
curvature lines are natural generalizations of Wente constant mean curvature
tori, discovered in 1986. Wente tori are recovered in a limit case of our
formulas.
