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We present new real algebraic maps of non-positive codimensions withprescribed images whose boundaries consist of explicit non-singular realalgebraic hypersurfaces satisfying so-called "transversality" as follows.Explicit information on important real polynomials is given. Preimages areone-point sets or products of spheres. They are locally like so-called momentmaps.  Celebrated theory of Nash and Tognoli says that smooth closed manifolds are{\it non-singular} real algebraic manifolds and the zero sets of some realpolynomial maps. In general, we can approximate smooth functions or moregenerally, maps, by real algebraic ones. It is in general difficult to haveexplicit examples. We have constructed maps of a specific class of the presentclass containing the canonical projections of the unit spheres previously wherepreimages are one-point sets or spheres.  We also present explicit families of functions represented as compositions ofsuch maps with the canonical projections.

Reconstructing real algebraic maps locally like moment-maps with  prescribed images and compositions with the canonical projections to the  $1$-dimensional real affine space