Ground-state of fractional and integral quantum Hall systems at $\nu   \leq 1$ and it excitations
  Many-body variational ground-state wave function of two-dimensional electron system (2DES), localized in the main strip (MS)$L_{x}^{\square} \times L_{y}$ of the finite width $L_{x}^{\square}=\sqrt{2 \pi m} \ell_{0}$ (and the periodic boundary condition (PBC) imposed along $x-$direction), is presented at the fractional and the integral filling factors $\nu=1/m$ for two different ion backgrounds: microscopical uniform ion background (UIB) and classical ion jellium background (IJB); $\ell_{0}$ is the magnetic length, $m=2\ell+1$ and $\ell=0, 1, 2,...$. It is shown that the ground-state and the lowest excited-state can correspond to partial crystal-like correlation order among $N$ electrons of the main region (MR) $L_{x} \times L_{y}$; then the study of 2DES of $N$ electrons within MR is exactly reduced to the treatment of 2DES of $\tilde{N}=N L_{x}^{\square}/L_{x}$ electrons localized within MS, with PBC along $x$. The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the $y-$ direction, with the period $L_{x}^{\square}/m$, and independent of $x$. For IJB, at $m=3, 5$, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state (the Laughlin model uses IJB); the same holds at $m=1$. Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from $m$ strongly correlated quasielectrons (quasiholes) of the charge $e/m$ ($-e/m$. Quantized Hall conductance $\sigma_{H}=e^{2}/(2 m \pi \hbar)$ is obtained. The theory is in good agreement with experiments. 
O. G. Balev