  We give a new proof for the local existence of a smooth isometric embedding
of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian
curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the
sophisticated arguments via microlocal analysis used in earlier proofs.
  In Part 1, we introduce a new type of system of partial differential
equations, which is not one of the standard types (elliptic, hyperbolic,
parabolic) but satisfies a property called strong symmetric positivity. Such a
PDE system is a generalization of and has properties similar to a system of
ordinary differential equations with a regular singular point. A local
existence theorem is then established by using a novel local-to-global-to-local
approach. In Part 2, we apply this theorem to prove the local existence result
for isometric embeddings.
