  We address here a few classical lattice--spin models, involving $n-$component
unit vectors ($n=2,3$), associated with a $D-$dimensional lattice
$\mathbb{Z}^D,\,D=1,2$, and interacting via a pair potential restricted to
nearest neighbours and being isotropic in spin space, i.e. defined by a
function of the scalar product between the interacting spins. When the
potential involves a continuous function of the scalar product, the
Mermin--Wagner theorem and its generalizations exclude orientational order at
all finite temperatures in the thermodynamic limit, and exclude phase
transitions at finite temperatures when $D=1$; on the other hand, we have
considered here some comparatively simple functions of the scalar product which
are bounded from below, diverge to $+\infty$ for certain mutual orientations,
and are continuous almost everywhere with integrable singularities. Exact
solutions are presented for $D=1$, showing absence of phase transitions and
absence of orientational order at all finite temperatures in the thermodynamic
limit; for $D=2$, and in the absence of more stringent mathematical results,
extensive simulations carried out on some of them point to the absence of
orientational order at all finite temperatures, and suggest the existence of a
Berezinski\v\i-Kosterlitz-Thouless transition.
