  $\mathcal{HH}$ spaces of type $[\textrm{N}] \otimes [\textrm{N}]$ with
twisting congruence of null geodesics defined by the 4-fold undotted and dotted
Penrose spinors are investigated. It is assumed that these spaces admit two
homothetic symmetries. The general form of the homothetic vector fields are
found. New coordinates are introduced which enable us to reduce the
$\mathcal{HH}$ system of PDEs to one ODE on one holomorphic function. In a
special case this is a second-order ODE and its general solution is explicitly
given. In the generic case one gets rather involved fifth-order ODE.
