I will consider autonomous systems. Noether’s theorem is easier in the Hamiltonian setting. Let $H$ be the Hamiltonian on a $2n$ dimensional phase space with canonical coordinates $x_i,p_i$ and $K$ the kinetic energy:
$$
K=\sum \frac12p_i^2
$$
Conservation of energy means that in terms of Poisson brackets:
$$
\{H,K\}=0
$$
This means that $H$ is invariant by the symmetry generated by $K$:
$$
\dot x_i=\{x_i,K\}=p_i \quad \dot p_i=\{p_i,K\}=0
$$
i.e. it is invariant by your transvections:
$$
x_i\to x_i+p_is\quad p_i\to p_i
$$
Physically, this symmetry is just ballistic motion. Thus your Hamiltonian is invariant by ballistic motion:
$$
H(x_i,p_i) = H(x_i+sp_i,p_i)
$$
It is not hard to find all of such systems. A simple way is choose a transverse hypersurface to the flow lines. A suitable candidate is the hyperplane:
$$
\sum x_i\frac{p_i}{\sqrt{2K}}=0
$$
Thus such Hamiltonians are of the form for any function $h$ defined on the hyperplane:
$$
H=h\left(x_i-p_i\sum_j x_j\frac{p_j}{\sqrt{2K}},p_i\right)
$$
Geometrically, this means that $H$ is a function of $p$ and the normal components of $x$ to $p$.
For physical applications, this is rarely the case. In non relativistic mechanics, you typically have $H = K+V$ with $V$ a position dependent potential. The only $V$ compatible with this symmetry is a constant, so conservation of kinetic energy in this case is trivial since it is just the Hamiltonian. Even adding magnetism will not help.
