  This paper presents a new algorithm based on interval methods for rigorously
constructing inner estimates of feasible parameter regions together with
enclosures of the solution set for parameter-dependent systems of nonlinear
equations in low (parameter) dimensions. The proposed method allows to
explicitly construct feasible parameter sets around a regular parameter value,
and to rigorously enclose a particular solution curve (resp. manifold) by a
union of inclusion regions, simultaneously. The method is based on the
calculation of inclusion and exclusion regions for zeros of square nonlinear
systems of equations. Starting from an approximate solution at a fixed set $p$
of parameters, the new method provides an algorithmic concept on how to
construct a box $\mathbf{s}$ around $p$ such that for each element $s\in
\mathbf{s}$ in the box the existence of a solution can be proved within certain
error bounds.
