  We construct explicit analytic solutions of the $SU(N)$-Skyrme model (for
generic $N$) suitable to describe different phases of nuclear pasta at finite
volume in $(3+1)$ dimensions. The first type are crystals of Baryonic tubes
(nuclear spaghetti) while the second type are smooth Baryonic layers (nuclear
lasagna). Both, the ansatz for the spaghetti and the ansatz for the lasagna
phases, reduce the complete set of Skyrme field equations to just one
integrable equation for the profile within sectors of arbitrary high
topological charge. We compute explicitly the total energy of both
configurations in terms of the flavor number, the density and the Baryonic
charge. Remarkably, our analytic results allow to compare explicitly the
physical properties of nuclear spaghetti and lasagna phases. Our construction
shows explicitly that, at lower densities, configurations with $N=2$ light
flavors are favored while, at higher densities, configurations with $N=3$ are
favored. Our construction also proves that in the high density regime (but
still well within the range of validity of the Skyrme model) the lasagna
configurations are favored while at low density the spaghetti configurations
are favored. Moreover, the integrability property of the present configurations
is not spoiled by the inclusion of the subleading corrections to the Skyrme
model arising in the 't Hooft expansion. Finally, we briefly discuss the large
$N$ limit of our configurations.
