  We describe the set of all $(3,1)$-rational functions given on the set of
complex $p$-adic field $\mathbb C_p$ and having a unique fixed point. We study
$p$-adic dynamical systems generated by such $(3,1)$-rational functions and
show that the fixed point is indifferent and therefore the convergence of the
trajectories is not the typical case for the dynamical systems. We obtain
Siegel disks of these dynamical systems. Moreover an upper bound for the set of
limit points of each trajectory is given. For each $(3,1)$-rational function on
$\mathbb C_p$ there is a point $\hat x=\hat x(f)\in \mathbb C_p$ which is zero
in its denominator. We give explicit formulas of radii of spheres (with the
center at the fixed point) containing some points that the trajectories (under
actions of $f$) of the points after a finite step come to $\hat x$. For a class
of $(3,1)$-rational functions defined on the set of $p$-adic numbers $\mathbb
Q_p$ we study ergodicity properties of the corresponding dynamical systems. We
show that if $p\geq 3$ then the $p$-adic dynamical system reduced on each
invariant sphere is not ergodic with respect to Haar measure. For $p=2$, under
some conditions we prove non ergodicity and show that there exists a sphere on
which the dynamical system is ergodic. Finally, we give a characterization of
periodic orbits and some uniformly local properties of the $(3.1)-$rational
functions.
