  We investigate the level spacing distribution for three ensembles of real
symmetric matrices having additional structural constraint to reduce the number
of independent entries to only $ (n+1)/2 $ in contrast to the $ n(n+1)/2 $ for
a real symmetric matrix of size $ n \times n $. We derive all the results
analytically exactly for the $ 3\times 3 $ matrices and show that spacing
distribution display a range of behaviour based on the structural constraint.
The spacing distribution of the ensemble of reverse circulant matrices with
additional zeros is found to fall slower than exponential for larger spacing
while that of symmetric circulant matrices has poisson spacings. The
palindromic symmetric toeplitz matrices on the other hand show level repulsion
but the distribution is significantly different from Wigner. The behaviour of
spacings for all the three ensembles clearly show the departure from universal
result of Wigner distribution for real symmetric matrices. The deviation from
universality continues in large $ n $ cases as well, which we study
numerically.
